/jp/ - Otaku Culture

Hey anons, relearning some nihongo and came across this fucking unexplained thing while going through Genki I again (pic related).

>(verb stem) + ~ってきます

I reread the whole book ever since the introduction of Te form up to this dialogue and nowhere does this form appear. I understand it's talking about the "future", but how does it work and what does it mean exactly?

What would the diffference be between that example and たけしさんにおみやげを買います。?