I don't think this is anything special about primes or e - if you replace prime(n+1) with just (n+1) itself you get the same sort of patterns. But it is something to do with approximations of irrationals by rationals - you might want to look into continued fractions. Try replacing e with a rational number a/b (say 8/3 or 11/4); then you get b horizontal-ish lines, corresponding to the different remainders of n when divided by b. So the pattern you get with pi isn't "as good" because pi is famously close to 22/7.
