Firstly why the script fails to rotate the letters by one, if I remove either $_++ (I've tested several combinations with and operator, and that didn't work either)?

$x++ returns $x, and increments $x in-place. So the first $_++ increments $_, and the the second instance of $_++ only has the function to return $_; the actual increment it does is lost. You can achieve the same thing with map { $_++; $_ }.

Also that might be just me not fully understanding the implications of the eval string hack

Afaict it does nothing (and I don't see any changed behavior if I remove it).

Also note that your code turns z into aa, a feature not desired for rot13.

but why won't $_++ for 1..13 or $_ += 13 work?

$_++ for 1..13 doesn't work because foreach has its own $_ variable, which is set to the item it currently iterates over. Something like $x++ for 1..13 should work.

 $_ += 13 assumes numeric context, ie it lacks the magic that ++ possesses.

Cheers, that pretty much answers everything. To the eval, I thought that in order to increment a string with ++, you had to eval it first; although I don't quite remember why that was...

As moritz already pointed out, this just relies on the way that the incrementer works on characters. Your code can be reduced a little further by using the preincrementor within the map, although it won't solve the problem with 'z' becoming 'aa'. But tighter still would be to just use a for:

#!/usr/bin/perl
use 5.12.0;
my @rot = qw(a v c g h t);
$_++ for @rot;
say @rot;
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- Miller