The きました comes from the verb くる. It means 'to come' or something like that, but of course used like this no one is actually moving. In this sentence すく is in the te form. One of the uses of the te form is to connect stuff together, so すく(to be hungry) is connecting to きました(the past tense of くる), to literally mean something like 'i have come to be hungry at this point in time' or 'I'm just starting to get hungry (so lets look for something to eat') etc, of course in English we just say 'I'm getting hungry' like yoshi said. You can use the grammar 'te kuru' / 'te iku' with any verb you like (although it can be strange sometimes) eg. ふとる　(to grow fat) - ふとってきましたね。 OK. ふとっていく。　＝　Strange.... but the grammar is ok. Its hard to explain but many textbooks and websites have example sentences using the te kuru / te iku structure. The best way is just to listen to natives speaking and studying lots of example sentences. Good luck!

Although yoshi-san and Jamie-san have already given good answers, let me add some just for your information:) -てくる is used to express that some "change" has just emerged. For example, 「おなかがすいてくる(start feeling hungry.)」「雨（あめ）がふってくる(start raining.)」「悲しくなってくる(start feeling sad)」. In this meaning, -ていく doesn't work. Also, -てくる can mean something change from the past to now(or at the point of speaker's view). The opposite is -ていく; it means something changes from now to the future. In other word, for -てくる, the speaker is looking at the past. For -ていく, the speaker is looking at the future. 「最近ふとってきた><」(I've been getting fat recently.) 「私、だんだんふとっていくみたい・・・。」 (It seems I'm going to keep getting fat gradually...)

I think.... おなかがすいてきました means "I'm getting hungry" This is a kind of vague expression. Not very much but you feel little hugry or feel like having something to eat. おなかがすいた means "I'm hungry" You are sure that you are hungry. You want to have something to eat. I hope this helps you to understand the meaning(: