Study the concept of Derangements from a good resource.
Derangement is basically the opposite of arrangement. In simple words, given $n$ objects and $n$ locations, how many ways can you arrange them such that all $n$ are in the wrong spots.
There is an accurate formula for this, but it can be approximated with $\frac{n!}{e}$
For $n = 5$, derangements $= \frac{5!}{e} \approx 44$
So there are $44$ ways to put all $5$ letters into the wrong envelopes, and $5!$ total ways of arranging the $5$ letters
$\therefore$ probability that all $5$ letters are in wrong envelopes $= \frac{44}{5!} = \frac{44}{120}$
$\therefore$ probability that at least one letter is in the right envelope $= 1 -\frac{44}{120} = \frac{76}{120} = \frac{19}{30} $