A language is recursive if a Turing Machine decides it: On input w, the TM always halts, and accepts if w is in the language, and rejects if w is not in the language.
So to show that the language is recursive, you just need to show that your Turing Machine gives the correct answer on every input string (accepts if # of a's equals # of b's, rejects otherwise). That will also show that it halts on all inputs, so you're done.
As Hendrik says in the comment, a Turing Machine should have both an accept state and a reject state. (That's what it means to accept/reject -- to enter the appropriate state.)
So first, you'll need to make this change.
Then, to prove your construction correct, you use the following type of proof. First, suppose I have a string $w$ that does have an equal number of a's and b's. Then I will show that, when I feed $w$ into my TM, it halts in the accept state. Second, suppose I have a string $x$ that does not have an equal number of a's and b's. Then I will show that, when I feed $x$ into my TM, it halts in the reject state.
You should convince yourself that this proof accomplishes what I said is required above!
But note your TM is not quite there yet: Consider what will happen on input string "bbb"!