Get in touch with the Realtor and ask how much he/she would charge to bring
you the renter. I don't think the Realtor gave you the name of the prospective
renter or your question would be different.
Negotiate the price. You can advertise the home yourself on places like
Craigs List. Or you can ask that Realtor to advertise your home on the local
mls. Expect to pay anywhere from a few hundred dollars up to one month's rent.
If you rented it out yourself, without the Realtor being involved and without having a listing agreement signed with the Realtor, you'd owe nothing.
Hope that helps.
Unless you have a signed agreement, the agent would be hard pressed to demand a fee. You may however want to consider the value of the service they provided and discuss the subject with them. If they did it expecting nothing, you may have a good friend in the business worth referring people to. If they were hopeful of some compensation, realize that if you didn't have this contact, your rental may be vacant a while longer. Is it worth something to you to have a good tenant in the home sooner rather than later?