Firstly as S K mentions in the other answer, the Chidambara Rahasya is just a vast expanse of emptiness. But there is more to what meets the eye.
Shiva is said to be present across the universe in 8 forms. Pushpadanata, in his famous Shiva Mahima Stotra (you can listen to a rendition of that on YouTube at 18:55), enumerates them as:
(You can learn more about the ashtamurthi here What are the eight forms (Ashtamurti) of Lord Shiva?).
These eight forms of Shiva represent one of the forces of nature respectively. Similarly, there is one temple present for each of them. You can get the mapping of the Ashtamurthi - Form of Nature - Temple from the Sanathan Dharma Project website.
Now let's focus just on one of them, which is Bheema. From the above link:
Bheema: - Akaasha Linga, Chidambaram, Tamil Nadu. This Kshetra is on the banks of Cauvery. We don’t see any Murthy in the temple Garbha Gruha. The puranas speak of this Kshetra very highly. No one can see the Lord’s Murthy, except the highest spiritual souls. There is a space in the Garbha Gruha and many Abharanas are decorated and the devotees assume the Lord is seated there. A very beautiful Nataraja murthy is in outer Garbha Gruha for worship and for the satisfaction of the devotees.
The Chidambaram temple is representative of the Akasha Murthi form of Shiva, called Bheema. He represents the space and outer cosmos. The etymology of the word Chidambaram is:
The name of the town of this shrine, Chidambaram comes from the Tamil word Chitrambalam (also spelled Chithambalam) meaning "wisdom atmosphere". The roots are citt or chitthu means consciousness or wisdom while and ampalam means "atmosphere"
Chidambara Rahasya, refers to the fact that there is no visible deity in the sanctum sanctorium. Instead there's just an empty place. This is to symbolize that the Shiva there represents the Space - or Akasha. The golden chains hanging from the top, are shaped in the form of Bilva leaves (in order to decorate the invisible Shiva idol). A more comprehensive account of the temple and the rahasya is given in the book Temples of South India by VVS Reddy.
In the chit-sabha to the right side of Nataraja there is the proverbial secret of Chidambaram (Chidambara rahasya), where several strings of golden bilva leaves hang in front of a curtain, behind which is empty space, but it is said to be akasa-linga, one of the panchabhuta (five) lingas. Akasa in other words is nothingness, or void, and thus, the linga is said to be there, but it not visible-actually nothing exists there but, people believe that there is actually the akasa-linga. This make-believe process is the essence of Chidambara-rahasya. Here it has been provided to worship the empty space itself as god.
The book The Madras Presidency: With Mysore, Coorg and the Associated States by Edgar Thurston, also provides an explanation about what the Chidambara Rahasya is:
At Chidambaram the emblem of the god is the ether linga, which has no actual existence, but is represented by an empty space in the holy of holies called the akasa or ether linga, wherein lies the so-called Chidambara rahasya, or secret of worship at Chidambaram.
As S K mentioned earlier, the priest will lift the curtain and show you the space behind the curtain. The curtain itself symbolizes Maya. People need to lift the shroud of Maya, to stare into that emptiness, where you can see Shiva in the form of Bheema. From the temple website:
Since ancient times, it is believed that this is the place where Lord Shiva and Parvathi are present, but are invisible to the naked eyes of normal people. In the Chidambaram temple of Lord Nataraja, Chidambara Ragasiyam is hidden by a curtain (Maya). Darshan of Chidambara Ragasiyam is possible only when priests open the curtain (or Maya) for special poojas. People who are privileged to have a darshan of Chidambara Ragasiyam can merely see golden vilva leaves (Aegle Marmelos) signifying the presence of Lord Shiva and Parvathi in front of them. It is also believed that devout saints can see the Gods in their physical form, but no such cases have been officially reported.
There's a lot more story about the temple, including that of Nataraja and his cosmic dance, which doesn't come under the scope of this answer