How does lemon juice “cook” meat?

The Oxford English Dictionary defines to cook as: Prepare (food, a dish, or a meal) by mixing, combining, and heating the ingredients. ^(Ref)

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.

It is important to understand that from a purely cooking perspective the two things are not the same: “cooking” with lemon (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!

From Wikipedia – Denaturation
When food is cooked, some of its proteins become denatured. This is why boiled eggs become hard and cooked meat becomes firm.

A classic example of denaturing in proteins comes from egg whites, which are typically largely egg albumins in water. Fresh from the eggs, egg whites are transparent and liquid. Cooking the thermally unstable whites turns them opaque, forming an interconnected solid mass. The same transformation can be effected with a denaturing chemical. Pouring egg whites into a beaker of acetone will also turn egg whites translucent and solid. The skin that forms on curdled milk is another common example of denatured protein. The cold appetiser known as ceviche is prepared by chemically “cooking” raw fish and shellfish in an acidic citrus marinade, without heat.