How does lemon juice "cook" meat?
Let's start with a definition of cooking.

The Oxford English Dictionary defines to cook as:


  To prepare or make ready (food); to make fit for eating by due application of heat, as by boiling, baking, roasting, broiling, etc. 


However, from a chemical point of view, what happens when you are cooking meat is that you are using heat, amongst other things, to denature the proteins of the meat.
Denaturation can also be achieved by other means, such as marinating the meat in an acid, such as citric acid (contain in lemon juice) or acetic acid (contained in vinegar).

You can also use other agents to induce denaturation, but they are not always fit for eating.

However, it is important to understand that from a purely cooking perspective the two things are not the same: "cooking" with lemon (I guess it is fine to use the verb to cook as long as you put it in quotes), gives a completely different result than cooking with heat. This is because other chemical reactions happen during application of heat and because, obviously, you do not end up with an hot product!