Learn Hardcore Hebrew: הראיה אצל השוכר, ואילו הערבות הבנקאית אצל המשכיר. - The renter has the evidence, whereas the landlord has the bank guarantee.

That is one of its basic meanings, but in sentences about people it often extends to with, in the possession of, or on the side of someone.

So in this sentence:

הראיה אצל השוכר = the proof / evidentiary burden is with the tenant
הערבות הבנקאית אצל המשכיר = the bank guarantee is with the landlord, in the landlord’s possession

So אצל is not always just physical location. It can also indicate who holds something or who it rests with.

Why is there no Hebrew word for is in this sentence?

Because Hebrew usually leaves out the verb to be in the present tense in sentences like this.

So:

הראיה אצל השוכר literally looks like the proof with the tenant
but it means the proof is with the tenant

This is completely normal Hebrew.

If you changed the tense, you would use a form of היה:

Past: הראיה הייתה אצל השוכר
Future: הראיה תהיה אצל השוכר

What does ואילו mean here?