What is meaning, anyway?  In English, it's easy to see that grammatical correctness and meaningfulness are not the same there.  Look at the following two sentences:
 

Both sentences are grammatically correct. But the first is meaningful, the second not.  The grammer of English tells us how to put words together, but it doesn't tell us how to make meaninful assertions!  What makes a sentence meaningful in English? That's a hard question, and one we're not going to even try to answer. Our semantics has a much more modest scope.

In PL we're concerned with simple sentences which are true or false, and with compound sentences built out of simple sentences with operators.  So the only aspect of the meaning of simple sentences which concern us is their truth value.  Simple sentences are either true or false.  Compound sentences are built out of simple ones. So for compound sentences, the operators which build up compound sentences from simple ones are the object of our attention. What do the operators mean?  That is, what happens to the truth-value of  compound sentences constructed from simple sentences? The semantics of PL tells us exactly this, for any possible compound sentence.