Size of $E$ over $\mathbb{F}_p$ contains $p+1$ points I am struggling to prove this claim:

I proved that the map $x\mapsto x^3+1$ is a bijection from $\mathbb{F}_p$ to itself if we have that $p\equiv 2\bmod{3}$. We have to use this fact to prove that the elliptic curve $E:y^2=x^3+1$ over $\mathbb{F}_p$ contains precisely $p+1$ points. 

I don't understand how many points will be on the curve. For each value of $x\in\{0,1,2,\ldots,p-1\}$ we will get a distinct result after computing $x^3+1$. But how does that tell me how many points will be on the curve?

For each point it can be paired with either two or zero $y$ values, right? But I don't know how many squares there are that are less than $p$.

I ask either for help explaining this, or ideally resources I could use to understand the theory behind this so that I can solve it myself.
For every $y \in \mathbf F_p$, there is a unique $x \in \mathbf F_p$ such that $(x,y)$ is on the curve, namely $x = \phi^{-1}(y^2)$, where $\phi : x \mapsto x^3+1$. Adding the point at infinity, that gives $p+1$ points.