Some propositions are true, and some are false. However, not all true propositions are obviously true, and not all false propositions are obviously false! For example, can you tell which of the following statements are true, and which are false?
Some of the statements (like the first few ones) may be intuitively true or intuitively false. But some of them aren’t obvious at all. For example, I can tell you that the fourth statement is true, and the fifth statement is false, but these are not obvious at all. Furthermore, no one even knows whether the sixth problem is true! It’s still an open problem in mathematics, and it's called Goldbach's conjecture.
So given this, how can we say which propositions are true and which are false? How do we get to know the truth? This is where proofs come in. A proof is just an argument showing that some statement is true or false, based on logic and what we already know to be true.
A great introduction to proofs is Chapter 2 of MIT 6.042. I encourage you to read it.