Let us consider what happens when we try to expand $ (x+h)^n = (x+h)(x+h)\cdots (x+h) .$
To expand the RHS, we must pick either $x$ or $h$ from each bracket, and multiply them together to produce a term. The full expansion consists of the sum of the terms produced over all combinations of us picking $x$ or $h$ from each bracket.
What if we pick no $h$'s from any of the brackets? Then we pick $x$ every time, and the term produced is $x^{n}.$
Now, what if we pick $h$ exactly from the 1-st bracket? Then we must pick $x$ from the other $n-1$ brackets, so the term produced is $hx^{n-1}.$ Now if we pick $h$ exactly once, but from the 2nd bracket, same term is produced. We can pick our single $h$ from any of the $n$ brackets, so that means all the terms produced by picking exactly one $h$ sum to $nhx^{n-1}.$
Now, we've considered what happens if we pick 0 $h$'s (we get $x^n$) and 1 $h$ (we get $nhx^{n-1}$. Everything else must involve picking $h$ at least twice, and if we pick $h$ twice the term produced has at least a factor of $h^2$ in it, explaining the "stuff involving $h^2$ as a factor" term.