Why does かいに きた mean "buy"?

Essentially, I've been playing Pokemon X on the Japanese language setting. In a shop, the options are:

かいに きた
うりに きた
なんでも ないです

I've figure out that the last line translates to "not anything," but I've searched everywhere for a translation of the first two lines. I did watch a lets play of the game in English and the top was translated to "Buy," while the second line was translated to sell. But I have no clue as to how those two lines mean this. The closest translation to that is google, but its very bad at explaining how. I did find out that うりに is the beginning of the word うりにだす, which means to offer for sale. But I am still highly confused.

Maybe this will help?
買いに来た
売りに来た

The beginning kanji definitely help. This makes me think that に is actually a particle. (I have no idea why that didn't cross my mind.) However, 来た still puzzles me. No matter how I look at it, its the ta-form of くる, meaning to come. Is this some sort of expression? You come to sell, or come to buy?

That's right. In general, "<stem of verb X> に <verb Y>" means "Y with the purpose of X", so e.g. "I came here in order to buy something". The verb stem is simply the -masu form without the -masu.

Also, なんでもない is a common expression meaning "It's nothing", "Nevermind".

Thank you. I understand now. I seems so simple once you understand it.