Can someone explain to me how this is a proof?

Question

I honestly don't understand how this proves anything.

Answer 1

How easy this is to state depends on what definition of recursive you are using, I'll use the second one on Wikipedia:

A recursive language is a formal language for which there exists a Turing machine that, when presented with any finite input string, halts and accept if the string is in the language, and halts and rejects otherwise. The Turing machine always halts: it is known as a decider and is said to decide the recursive language.

So what the picture is saying is that as $w \in L \iff w \not\in \overline{L}$ if $L$ is recursive we can construct a decider for $\overline{L}$ by simply reversing the outputs of the decider for $L$. Here the box with $M$ in is meant to represent the decider for $L$ and the larger box is the new Turing machine that decides $\overline{L}$

This sort of diagrammatic argument can be hard to get used to but in situations like this it is fairly common and once you are used to them they are quicker to write down than words!