I got introduced with roughly the following explanation.
Imagine a secure room with a great number of
piggy banks. In this room there are a great number of cameras keeping watch over everything that is happening in the room. Anyone is free to place a camera in the room and keep watch for themselves, and anyone is free to place a piggy bank in the room to keep their money in.
Each piggy bank has a public name which anyone can see, and a private key which is given to whoever placed that particular piggy bank in the room. If you want someone to put money in your piggy bank, you can just tell them the public name of your piggy bank and they will be able to do so.
Whenever YOU want to take coins out of your piggy bank and put it in one of the other piggy banks (which belong to other people) imagine the following: Walking into the room, using your private key to open your piggy bank, and placing as many coins as you want in one of the other piggy banks.
The cameras in the room will not be able to see your identity or your private key. The cameras only see that things are done in the proper way; that piggy bank X was opened in the right way, and that a certain amount of coins from piggy bank X was put in piggy bank Y.
What are cameras? Cameras are (open source) programs that any computer can run when connected to the internet. Why do people contribute cameras? Each user is rewarded with a small amount of new coins to the extent that they help keep watch of the room. And the more people help by keeping watch, the more secure the room will be.