This is what I have written: is it this correct?
We can prove this by simply saying that if $p$ is a prime number, then the only divisors of $p$ are $p$ and $1$. So we have $\sigma(p)=p+1$.
Conversely, if we have $\sigma(N)=N+1$, this tells us the only divisors of $N$ are $N$ and $1$, which makes it a prime number.
Yes, this is fine.
When writing out a proof at this level I might spell out more explicitly the fact that 1 and N always divide N so that $\sigma(N)\geq N+1$ in general rather than just saying "This tells is the only divisors are 1 and N" without explanation.
Once you move to proofs where details like that aren't the crux it's okay (and recommended) to leave them out.