Aegle Marmelos or Bel Leaves are used in the rituals related to Lord Shiva. What is the significance of these leaves in Hinduism?
The reason that the leaves of the Bilva tree are used in the worship of Shiva is that the Bilva tree emerged from the sweat of Shiva's wife Parvati, as described in the this excerpt from the Nagara Khanda of the Skanda Purana:
The splendid Daughter of the Mountain; the goddess [Parvati] became tired due to her sports. Drops of sweat appeared on her forehead ... and a drop fell on the ground. It became a great tree on the excellent mountain Mandara.... Parvati said, "Since this tree pierced through the ground and rose up near me, let it be named Bilva. If anyone approaches this tree devoutly and collects the leaves, he shall certainly become a king of the earth. If anyone with perfect faith worships me with its leaves, he will realize whatever desire he may cherish. If a person, after seeing the leaves of Bilva, at least maintains faith in the rite of adoration, I shall undoubtedly be the bestowed of wealth on him. If anyone makes his mind to eat the tip of the leaves, thousands of his sins perish automatically. If a man places the tip of the leaf on his head, no torture of Yama will give him pain." ...
This Bilva tree is very excellent, sacred and destructive of sins. There is no doubt about this that Goddess Girija resides at its root. It is remembered that Dakshayani is present on its stem, Maheshwari on its branches, Goddess Parvati on its leaves and Katyayani on its fruit. Gauri has been mentioned as present on its bark, Aparna at the middle of the bark, Durga in the flower and Uma in the branches and twigs. At the behest of Girija, nine crores of Shaktis have stationed themselves on its thorns for the sake of protecting living beings.
Those are all names of different forms of the goddess Parvati. Since every part of the Bilva tree is suffused with some form of Parvati, it is the ideal article to be used in worship of Shiva, just as Tulasi is ideal for worshipped Vishnu as it's associated with Vishnu's wife Lakshmi.
There is a stotra (hymn) called Bilwastakam, that describes the spiritual benefits of using even one bel leaf for lord Shiva. For example, the first verse says:
tridalaṃ triguṇākāraṃ trinetraṃ ca triyāyuddham
trijanmapāpasaṃhāraṃ ekabilvaṃ śivārpaṇam
Offering one three petaled bel leaf which is like the three qualities, which is like the three eyes of Shiva, which is like the trident weapon of Shiva, destroys sins of three births.
So basically, the bel leaves used in Shiva puja signify the three eyes of Shiva and also His three pointed trident (or three weapons). Hence, bel trees, being a favorite of shiva, are also considered to be a form of Shiva as well (mahādevasvarupoyaṃ bilvo [Shiva Puran 22.22]).
The bilva tree itself is so holy and auspicious that its worship or its significance is mentioned in many puranas and other scirputres at various instances. Here below is a narration of "greatness of bilva" under 22nd chapter in vidyesvarasamhita of shivapurana.
Lakshmyaascha stana utpannam Mahaadeva sadaa priyam, Bilva vriksham prayachchhaami eka bilvam Shivaarpanam. Darshanam bilva vrikshasya sparshanam paapanaashanam, Aghorapaapasamhaaram eka bilvam shivarpanam.
Born from the breasts of Goddess Lakshmi, the Bilva tree is ever dear to Mahadeva. So I ask this tree to offer a Bilva leaf to Lord Shiva. To have darshan of the Bilva tree, and to touch it, frees one from sin. The most terrible karma is destroyed when a Bilva leaf is offered to Lord Shiva.Sri Bilva Shtakam (v. 6–7)
The proportion of sattva component is more in bilva patra and hence it has more capacity to absorb and emit sattvik frequencies. This has various effects. One of them is the reduction of raja-tama particles present in the atmosphere. A sattvik leaf like bilva patra when brought in proximity of a person suffering from negative energy distress then the black energy present within him is reduced.
I give below various references as to why the leaves of the Bilva tree are used in the worship of Shiva:
According to the Agni Purana, on any auspicious day in Bhadra, Shiva should be worshipped with a daylong fast and the eating of bel leaves at night The Padma Purana and the Brahma Purana say that Shiva once hid in the bel to escape conquering demons .The Skanda Purana holds that the bel grew from Parvati's perspiration, which fell to the ground while she performed penance. It also says that the various incarnations of Parvati reside in each part of the tree.
In the hymn Bilvashtakam there is these verses
“Lakshmyaascha stana utpannam Mahaadeva sadaa priyam,Bilva vriksham prayachchhaami eka bilvam Shivaarpanam.”
“Darshanam bilva vrikshasya sparshanam paapanaashanam,Aghorapaapasamhaaram eka bilvam shivarpanam”.
The translation : Born from the heart of Goddess Lakshmi, the Bilva tree is ever dear to Mahadeva. So I ask this tree to offer one Bilva leaf to Lord Shiva.
Even if one sees the Bilva tree, and touches it, he is surely freed from sin. The most terrible karma is destroyed when a Bilva leaf is offered to Lord Shiva.
Brihaddharma Purana says Lakshmi prayed to Shiva every day and offered him 10,000 lotus buds. One day she fell short by two buds. Remembering that Vishnu had compared her breasts to lotus buds ,she cut one off and offered it humbly. Before she could cut the other, Shiva, pleased with her ,stopped her. Her cut breast became the fruit of the bel.
The importance of fasting and praying on Shivaratri night is told in this legend: One night Lord Shiva's devotee, a hunter, lost his way in the forest. While wandering at night, he heard a tiger growl. Frightened he scurried up the nearest tree that was a Bel tree. He spent the night up there, out of the reach of tigers. To keep himself awake, he kept plucking and dropping Bel leaves while chanting the name of Lord Shiva.The next morning, when he climbed down, he had unknowingly dropped thousands of leaves on a Shiva linga. The night-long worship pleased Lord Shiva who saved him from the tiger.
There are many other stories as to why leaves of the Bilva tree are used in worshipping Shiva - take your choice.