or putting it another way, it's simply the -te form of nai. but since it ends in i, like i-adjectives, it goes to -kute. other examples are omoshirokute, yasashikute, warukute.
so it's essentially joining the 2 bits of the sentence together in your case. "okane ga nai. komatteimasu." it makes sense, but just whack nai into -te form and we get one flowing sentence instead of 2 short separate sentences.