Learn Hardcore Korean: biga waseo gabangi jeojeosseoyo.

오다 means to come, but it is also used for weather phenomena like rain and snow. So 비가 오다 literally means rain comes, but naturally it means it rains.

When 오다 is connected with -아서/-어서, it becomes 와서:

오다 → 와서

So 비가 와서 means because it rained or it rained, so...

What does -아서 / -어서 mean in this sentence?

In this sentence, -아서/-어서 connects two clauses and shows a reason or cause.

비가 와서 = because it rained
가방이 젖었어요 = the bag got wet

So the whole sentence means:

It rained, so the bag got wet
Because it rained, the bag got wet

This ending can also sometimes just show sequence, but here the meaning is clearly cause and result.

Why is it 비가, not 비는?

가 is the subject marker, and here it simply identifies 비 (rain) as the subject of the clause.

비가 와서 = because rain came / because it rained

If you used 비는, it would sound more contrastive, like:

As for the rain, ...
The rain, though, ...

That is not the neutral feeling of this sentence. 비가 is the most natural choice here.