Let $E_i$ be the expected frequency and $O_i$ the observed frequency.
For example, if we role a six-sided die $120$ times then $E_i = 20$ for all $1 \le i \le 6$, while the $O_i$ will be however many times the number $i$ is rolled. Define
$$X^2 = \sum \frac{(O_i-E_i)^2}{E_i}$$
I don't see why the following statement might be true: "$X^2$ is approximated well by $\chi^2$ as long as none of the expected values $E_i$ fall below $5$.
At first, I thought that because the example with the die had $5$ degrees of freedom, this is why we needed $E_i \ge 5$. Although, I didn't see why that might be the case either! However, it seems that $E_i \ge 5$ is a condition for any example, no matter what the number of degrees of freedom.
Why is this?
The approximation is it is a sum of standard normal variables squared. When the counts are small a normal approximation to the distribution is not reasonable. 5 is just a rule of thumb. The exact test when the counts are small requires a lot of computation so is avoided when possible, although these days its not such a big problem.