Why do the event of interests have to be rare when using the poisson distribtution?
small p and n*p < roughly 7 are required according to our book without explaining why.
We went through the proof of deriving the poisson distribution from the binomial distribution as n approaches infinity but a need for n and p didn't come up anywhere.
You wrote, "We went through the proof of deriving the Poisson distribution from the binomial distribution as n approaches infinity." Consider what happens when $n$ and $p$ are fixed and don't satisfy these requirements.
For example, if $n=4$ and $p=1/4$ in a binomial distribution, then $P(X=0)=(3/4)^4=81/256$. Now, compare this to $P(X=0)$ for a Poisson distribution with $\mu=1$. They aren't very close. This isn't in disagreement the proof that you went over, because that proof tells you what happens in the limit as $n$ goes to infinity.