How exactly does a home rental agency work?
I have about 2-3 months left to live in my current house, since the owner is losing the place. I have no money saved for a move, but starting this month, I will begin saving my rent money…so it’s not really an issue.
The problem is, that i’ve noticed many houses use an agent for their rentals…How does a rental agency work?

If I understand your question correctly, the short answer is the agency is contracted by the owner as the "property manager". They will be the one to find a tenant, collect rents and make repairs. They will also act as a liason between owner and tenant. Those two should never have contact. They are typically paid a percentage of the monthly rent. For example, the owner may want to net $800 per month for the property. The agent would then charge $1000 per month rent and net a $200 per month income on that property.