てくる — Come Doing / Has Been Doing (Toward Now)

最近、寒くなってきた。

Saikin, samuku natte kita.

It's gotten cold lately (up to now).

なって + きた = has become — change built up from the past toward now

コーヒーを買ってきます。

Koohii o katte kimasu.

I'll go buy coffee and come back.

買って + きます = go, buy, and return (physical movement back toward here)

駅の前に新しい店が増えてきました。

Eki no mae ni atarashii mise ga fuete kimashita.

New shops have been increasing in front of the station (up to now).

増えて + きました = have increased — change built up from the past toward now