What does '/' mean? Also, why does '+' seem to do different things in the first and second expressions? I suppose '!n' means "array of integers ending at n" which seems like an odd choice since it's neither 'not' nor factorial.
You are correct in supposing that !n generates an array of integers from 0 to n-1. It is somewhat of an unusual choice for a symbol. I've always thought of it as a factorial 'on the other side'. I don't know if that's what was intended. The k language compresses many old APL functions into symbols like this. Some make sense, some make less sense. You are after all dependent on your keyboard for the range of symbols you can type. In APL the same function was indicated by a lowercase Greek iota.
'+' is being used in two different ways. The first way shows it being used in conjunction with an adverb, so '+/' is like a one-argument function that gets applied to the list at the right, but then '+' is inserted between each of the elements of the list because of '/'.
'\' is a very similar adverb. It's called 'scan', and it's like 'over', except intermediate output is produced. It can help illustrate what's going on.
!100
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ..
+/!100
4950
+\!100
0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210 231 253 276..
5 + 0 1 2
5 6 7
5 + !3
5 6 7
1 2 3 + 4 5 6
5 7 9
1 2 + 4 5 6
'length (an error)