I'll take a crack at explaining this without technical jargon.
Lets say you want to send a nasty letter to someone, but you'd rather not deliver the letter in person for fear that they might get angry with you.
You can ask a courier to take the letter from your house, and deliver it to the recipient, right? That works, but has the problem that the courier knows both identities, and an evil courier could snitch on you.
You could use two couriers, and instruct the first to pass it on to the second, and instruct the second to pass it on to the recipient. That almost works, because now you are the only person that knows both endpoints to the conversation.
Courier A knows you and
courier B, while
courier B knows
courier A and the recipient;
You --- A --- B --- Recipient
And just to decrease the chances of getting a bad circuit (when all couriers are evil and working together), ToR uses 3 couriers, so add a '
C' to that diagram in your head.
There is one problem remaining; how do we deliver the instructions to each courier without revealing our identity? If you wrote 'pass to B' on one piece of paper, you can give that to courier
A alongside your nasty letter, but any instructions you send along with it,
A can also read, and will therefore know the identities of everyone on the circuit.
ToR solves this final problem with repeated encryption, encrypting the message and instructions repeatedly, once for each courier. In terms of our analogy, this is what you do;
Put your nasty letter in a red box alongside a piece of paper saying 'pass to recipient'.
Lock the red box with the key belonging to courier C.
Put the red box inside a blue box alongside a piece of paper saying 'pass to C'.
Lock the blue box with the key belonging to courier B.
Put the blue box inside a green box alongside a piece of paper saying 'pass to B'.
Lock the green box with the key belonging to courier A and pass it to him.
If you follow this through, you will see that none of the couriers can ever have knowledge of the full circuit, and there is no way for the recipient to find out that you sent the nasty letter.