I think you've conveyed the general idea of the proof well enough, though the delivery could use some polishing and you are missing quite a bit of detail. I don't really blame you though since a proof by contradiction in this scenario is a bit unnatural.
Suppose for the sake of contradiction that $f(A) \cup f(B) \neq f(A\cup B)$. Then there either exists $x\in f(A) \cup f(B)$ such that $x\notin f(A\cup B)$ or $x\in f(A\cup B)$ such that $x\notin f(A) \cup f(B)$. We will handle these separately.
First suppose that $x\in f(A) \cup f(B)$ and $x\notin f(A\cup B)$. Then $x\in f(A)$ or $x\in f(B)$. We must then have $x\in f(A) \subseteq f(A\cup B)$ or $x\in f(B) \subseteq f(A\cup B)$. A contradiction either way.
Suppose then that $x\in f(A\cup B)$ and $x\notin f(A) \cup f(B)$. Then there exists $y \in A\cup B$ such that $f(y) = x$. We have $y\in A$ or $y\in B$. In either case, $f(y) \in f(A)\cup f(B)$ and we obtain a contradiction.
If you have any experience with mutual set inclusion proofs then you'll see that the above is essentially a convoluted way of writing mutual set inclusion. The contradiction part is completely unnecessary and adds nothing to the proof.
A bit of a remark on your last question. Certainly the statement has to be true for you to prove it. A proof is a certificate of correctness. But oftentimes you have no idea as to whether a given statement is true or false.
In textbook scenarios, you are generally given a problem - guaranteed to be true - for you to prove. More often than not, the statement is not only true but has an elementary and elegant solution (i.e. olympiads). This is in direct contrast to say, mathematical research, where not only are the statement given not guaranteed to be true, but you often struggle to find even an underlying framework to present your problem. Research is done often with simultaneous approaches in proving and finding counter-examples to any given problem.
The task of deciding the validity of a statement ultimately falls upon your intuition, and human intuition is developed more for hunting buffalo than for judging mathematical validity. So as a rule of thumb, never assume anything to be true unless you have a valid proof of the statement, there have been cases in history in which we have been burned quite badly by our intuition.