To see why $n+n+...+n=n^2$, consider this:
The multiplication of two whole numbers is equivalent to the addition of one of them with itself as many times as the value of the other one; for example, 3 multiplied by 4 (often said as "3 times 4") can be calculated by adding 4 copies of 3 together:
$3 \times 4 = 3 + 3 + 3 + 3 = 12$
Remember the explanation of multiplication you learned back in grade school! Ask yourself why we say 3 times 4... the wording is not arbitrary - we are literally adding 3, 4 times. Equivalently adding $n$ copies of $n$ is equivalent to saying $n$ times $n$, or $n^2$.