Learn Danish: Hun bliver hjemme, for hun er syg. - She is staying at home, because she is ill.

Yes, bliver often translates to "becomes" or is used for the future tense. However, in this context it means "stays". The phrase at blive hjemme is a very common fixed expression meaning "to stay at home".

Why does the sentence use hjemme instead of just hjem?

Danish distinguishes between movement and location. You use hjem when there is movement towards your home (like "going home"), but you add an -e to make it hjemme when you are already there. Since she is staying in place, we use the location form hjemme.

Could I use fordi instead of for here?

Yes, you could say fordi hun er syg. Both mean "because", but they act differently in the grammar. For is a coordinating conjunction, which links two independent sentences and keeps the normal word order. Fordi starts a dependent clause, which changes the word order if you add an adverb like ikke (for example, fordi hun ikke er syg versus for hun er ikke syg).