てくる details

write + (and…) + has been getting

kaite kuru

has been getting; starts happening; go do and come back

meaning: come to write

日本語にほんごのニュースが少すこしずつ分かっわかってくる。

nihongo no nyuusu ga sukoshi zutsu wakatte kuru.

Little by little, I come to understand the news in Japanese. neutral

ちょっとコンビニ行いってくるね。

chotto konbini itte kuru ne.

I'm just popping out to the convenience store. casual

Examples

change developing up to now — a shift arriving at the speaker's present, including onsets felt in real time

日本語にほんごのニュースが少すこしずつ分わかってきた。

nihongo no nyuusu ga sukoshi zutsu wakattekita.

Little by little, I came to understand the news in Japanese. neutral

なんか雨あめ降ふってきたね。

nanka ame futte kita ne.

Hey, it's starting to rain. casual

go, do, and come back — a round trip anchored at "here"

ちょっとコンビニ行いってくるね。

chotto konbini itte kuru ne.

I'm just popping out to the convenience store. casual

昼食ちゅうしょくを買かってまいります。

chuushoku o katte mairimasu.

I will go and buy lunch (and be right back). formal

始める
Both report an onset — 雨が降り始めた states the phase change objectively from outside, while 雨が降ってきた reports it arriving at the speaker and carries an "I can feel it now" immediacy 始める lacks.

ていくplain-antonym
The same aspectual machinery pointed opposite ways — ていく tracks change or motion heading away from the speaker's here-and-now (減っていく), てくる tracks it arriving at the here-and-now (減ってきた); deictic direction is the entire difference.

てみる
In the "go check" scene they nearly collide — ちょっと見てくる promises a round trip away and back with the answer; ちょっと見てみる promises only an attempt on the spot; てくる moves the body, てみる only tests the outcome.