This is the 321st most frequent Japanese word.
来る
"来る" means "to come."
In this sentence, '来る' is used as part of the phrase '降ってきた', implying the action of rain starting and coming into the scene, often used to describe phenomena starting or arriving.
雨が降ってきた。
It started to rain.
Here, '来る' conveys the meaning of arrival, as in the receipt of an expected item or information.
そのメールがやっと来た。
That email finally arrived.
Here, '来る' is used in its primary meaning of 'to come', indicating movement towards a specific location.