How can mathematical induction prove something?

I'll explain weak induction, which is probably what you're learning. Let's say you want to prove a statement for $n\geq N$. Say this statement is $P(n)$.

1) Basis step: Prove the statement for $n=N$, i.e. $P(N)$ is true.

2) Inductive step: Suppose $P(m)$ is true for some $m\geq N$. Then you use this assumption to prove that $P(m+1)$ is true.

How the two steps work is as follows. In the inductive step you should have proved $P(m+1)$ is true using $P(m)$ is true, without explicitly stating what $m$ is (in other words, you DON'T substitute it with a number, say 100 or something).

This means that no matter what $m$ is, it will always be the case that when $P(m)$ is true, then $P(m+1)$ is true, and this holds for all $m\geq N$.

Now in the basis step, you have proved $P(N)$ is true. By the preceding paragraph, this means $P(N+1)$ is true. Then $P(N+2)$ is true. Then $P(N+3)$ is true, and so on. So $P(n)$ is true for all $n\geq N$.