In the example above, the speaker got business cards from a few potential customers. She thinks that she might be able to sell to them. She's optimistic about those people becoming customers, so she says that the leads are "promising".
Here are some things that are often called "promising":
Jules had a promising future as a management consultant, but she left it all behind to start her own organic bakery.
He works with promising young athletes to improve their strength, flexibility, and endurance.
Researchers are currently working on a promising new treatment for breast cancer.