Given that the great Abhimanyu was an expert at Chakra-Vyuha (a.k.a Padma-Vyuha), why couldn’t he manage to come out of it?
Krishna was also an expert in Chakra-Vyuha at that time, so why didn't he rescue Abhimanyu?
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Vyuha generally means an arrangement. So chakra vyuha means the arrangement of soldiers in the form of a wheel. This was an old warfare technique like garuda vyuha, makara vyuh, mandala vyuh etc. Chakra vyuh being hard to penetrate Drona chose it on the thirteenth day of war when Abhimanyu got killed.
Abhimanyu failed to break it because he had not learnt how to break it. In fact, he had not learnt the technique to enter into it in the traditional way. While he was in his mother's womb Arjuna used to talk about war strategies to Subhadra and from that only he had learned the technique. However, while talking about how to breakout of a chakra vyuha Subhadra got asleep and Abhimanyu couldn't listen the technique. So he only knew how to break into a chakra vyuha, but not how to break out of it in case of any danger.
Why didn't Shri Krishna rescue Abhimanyu?
Yes, it is true that apart from Abhimanyu, two other people on the Pandav side who knew about chakra vyuha strategy were Shri Krishna and Arjuna. But as per their strategy, before forming the chakra vyuha, Susarman of the Kauravas challenged Arjuna for fighting and lured him away along with his chariot driver Shri Krishna to the southern part of the battle field. So both Arjuna and Krishna were unaware about what was going on with Abhimanyu. Had they been present there, then Arjuna would have certainly done something if not Shri Krishna, as He had given his word to not use weapons in the battle.
As the story goes in Mahabharata, in their bed time Arjun was narrating a very ingenious Yyuha to his wife Subhadra. Subhadra was at that time expecting and she had Abhimanyu in her womb.
Subhadra was all vigilant during the first part of the narration which involved entering the Vyuha, but during the latter part i.e. exiting she felt sleepy and couldn't pay proper attention to it. So, Abhimanyu who was in her womb couldn't listen to the second part.
Mahabharata: Abhimanyu and Chakra-vyuha has a somewhat alternate story,
When Abhimanyu was in his mother's womb, Sri Krishna used to take Subhadra on excursions. To humour her, Krishna used to relate many of his adventures to the pregnant Subhadra. On one such excursion Krishna was narrating his experience with the technique of Chakra-vyuha and how step-by-step the various circles could be penetrated. Chakra-vyuha was a military formation which was an effective form of defence. The army would be arranged in the form of a circular grid and would then challenge the enemy to break that grid. Nonetheless, it seems that Subhadra did not find this topic interesting and she soon fell asleep. However, someone else was interested in Sri Krishna's narration - the yet to be born Abhimanyu.
While Subhadra dozed off, Abhimanyu continued to carefully follow Sri Krishna's narrative of the Chakra-vyuha. But, after talking for some time and not receiving any response from Subhadra, Sri Krishna realised that she was savouring a sweet nap. Sri Krishna, who had at that time come up to the seventh step of the Chakra-vyuha, gave up his narration and returned with Subhadra to the palace.
As per the promise given to Pandavas and Kauravas about usage and distribution of his resources, he wasn't supposed to participate in the Mahabharata war, rather he was only supposed to be an advisor. Krishna with his power could have rescued Abhmanyu easily, but his promise prevented him to do so..
Sri Krishna's Kurushetra Lila explains this strory beautifully,
Sri Krishna was lying down, resting, when Duryodhana and Arjuna arrived. Duryodhana entered first, and sat near the head of Sri Krishna. He grabbed a chair and reclined. Arjuna did not sit on a chair. He stood with folded hands at the feet of Sri Krishna. After a while, Sri Krishna opened his eyes, and first, naturally, he could see only what was near his feet, not his head.
“Arjuna, how come you are here at this moment?” said Sri Krishna.
“No sir! I have come first,” said Duryodhana from behind.
“Oh! You have also come,” said Krishna.
Sri Krishna said, “You have come first, but I saw Arjuna first. Also, he is younger, you are the elder. Don’t you think it is proper for me to speak to the younger one first, especially as I saw him first?”
Then turning to Arjuna, Sri Krishna asked, “What made you come here?”
Arjuna replied, “Great Master, you know what is going to happen. War has become inevitable. We all want your help.”
Sri Krishna said, “What can I give you? I have two things. I have a large army called Narayani Sena; if you want it, you can take it. Otherwise I am here, but unarmed, doing nothing. I will merely sit and discuss with you. I will not take part in the war. If you want such a man as I am, take me. Or if you think this is not going to be of any utility to you, take the large army which will help you, as it is almost invincible.”
“I want you only, Master,” replied Arjuna.
Immediately Duryodhana retorted, “I want the army.”
“Take it,” said Sri Krishna.
Duryodhana left the place hurriedly, and declared to the Kuru family that he had already won victory in the war, that his victory was certain because of the invincible forces that he had received from Sri Krishna.
When Duryodhana left the place, Sri Krishna accosted Arjuna and said, “What a foolish person you are! Why did you not ask for the army? What good is it if I sit idle without doing anything for you? Why have you made this wrong choice? The other man took the good forces, and you are asking for me, who is as good as nothing.”
Arjuna replied, “Thou art all for me, great Master. I know you very well. Don’t try to deceive me by this query as to why I have chosen you.”
“Oh! You want to vie with me. Okay, all right. Do that,” said Krishna.
Then they both left.
Abhimanyu’s untimely death is attributed to his enmity with Lord Krishna in his previous birth. Abhimanyu in his previous birth was known as Abhikasura a demon. He was a friend of King Kans who left no stone unturned to kill his nephew Krishna. After Krishna killed Kans, Abhikasura vowed to take revenge on the former. Krishna, sensing the moves of the demon, wove a magic around Abhikasura, transformed him into an insect and captured him in a box. When Krishna’s sister Subhadra, after her wedding with Arjuna, accidentally opened the box, the insect got into her womb. Abhikasura was then born as Abhimanyu. Krishna knew this and he rather sadly, prepared the ground for the death of his nephew and finally got him entrapped in Padmavyuha. When Arjuna was mourning the death of his son and squarely blamed Krishna for it, it was then Krishna explained Abhikasura’s reincarnation as Abhimanyu and the necessity for him, as the Supreme Power, to incinerate the evil force.
Abhimanyu was actually Varchas, son of Soma or Chandradeva. Varchas incarnated as Abhimanyu but Soma wanted him to spend less time on earth only for 16 years so Soma wanted an encounter to take place between Abhimanyu and other Kuru warriors where he forces his opponents to retreat and he wanted Abhimanyu to die in that Chakravyuha.
And he who was known as the mighty Varchas, the son of Soma, became Abhimanyu of wonderful deeds, the son of Arjuna. And before his incarnation, O king, the god Soma had said these words to the celestials, 'I cannot give (part with) my son. He is dearer to me than life itself. Let this be the compact and let it be not transgressed. The destruction of the Asuras on earth is the work of the celestials, and, therefore, it is our work as well. Let this Varchas, therefore, go thither, but let him not stay there long. Nara, whose companion is Narayana, will be born as Indra's son and indeed, will be known as Arjuna, the mighty son of Pandu. This boy of mine shall be his son and become a mighty car-warrior in his boyhood. And let him, ye best of immortals, stay on earth for sixteen years. And when he attaineth to his sixteenth year, the battle shall take place in which all who are born of your portions shall achieve the destruction of mighty warriors. But a certain encounter shall take place without both Nara and Narayana (taking any part in it). And, indeed, your portions, ye celestials, shall fight, having made that disposition of the forces which is known by the name of the Chakravyuha. And my son shall compel all foes to retreat before him. The boy of mighty arms having penetrated the impenetrable array, shall range within it fearlessly and send a fourth part of the hostile force, in course of half a day, unto the regions of the king of the dead. Then when numberless heroes and mighty car-warriors will return to the charge towards the close of the day, my boy of mighty arms, shall reappear before me. And he shall beget one heroic son in his line, who shall continue the almost extinct Bharata race.'
Soma said reappear so that basically means that Soma wanted Abhimanyu to return to the Chandraloka who was actually Varchas by dying in that Chakravyuha and clearly tells that this encounter should happen without Nara and Narayana taking part in it which was Arjuna and Krishna.
Even the gods also agreed to it.
Hearing these words of Soma, the dwellers in heaven replied, 'So be it.' And then all together applauded and worshipped (Soma) the king of stars.
When Vishnu was about to take birth as Krishna to wipe off adharma, Varchas aided him in the task by becoming Abhimanyu and the gods accepted and even agreed Soma's conditions.
So this is the reason why Abhimanyu was trapped in Chakravyuha.
Chakravyuha: A winding, ever-rotating circular formation; considered impenetrable in the Mahabharata by all warriors except for Bhishma, Krishna, Arjuna, Drona, Parshuram and Pradyumna.
It is the most toughest vyuha or battle formation.
Dronacharya used this chakravyuha to capture Yudhisthira on the 13th day but Abhimanyu entered that vyuha and got trapped as many warriors attacked him simultaneously.