I hope that someone can explain this more accurately, but let me try some explanation.
According to 敬語の指針, the document explaining the use of honorifics in Japanese written by the Council for Cultural Affairs, there are two different kinds of what have been traditionally categorized as 謙譲語 (humble words). Most of them are solely used for actions/belongings of a speaker or someone considered to be on the side of a speaker, and those are called 謙譲語 I. However, the other humble words such as 参る and 申す can also be used for actions/belongings of something which is on neither the speaker’s side nor the audience’s side. In this usage, they play almost the same role as 丁寧語 (polite words). These humble words are called 謙譲語 II.
Therefore, in the example sentence, 参る is essentially used to make the sentence simply more polite than 雨が降ってきました.
To emphasize the difference between 謙譲語 I and 謙譲語 II, we cannot rephrase the example sentence as 雨がお降りしてきました because お…する forms 謙譲語 I from a verb.