A Question Based on Properties of Direct Product of Fields

The geometric explanation of the other post is fine, but here's an easy method to find it in $2$-space, and indeed $n$-space.

There are $p^n$ points in an $\mathbb{F}_p$-vector space of dimension $n$. There are therefore $p^n-1$ non-zero points. There are $p-1$ non-zero points on a line, and no two lines share non-zero points (because a line is all scalar multiples of a point). Thus there are $$ \frac{p^n-1}{p-1}$$ lines in $n$-dimensional space. If $n=2$, this yields $p+1$.