A "compelling" argument is one that's convincing. It makes you agree with what the person making the argument is trying to say. In the example above, the salesperson made a compelling argument that the couple should buy a more expensive car.
Other common combinations for "compelling" are:
- make a compelling case
Antonio makes a compelling case for sticking with our current suppliers.
- present compelling evidence
Most of the jury was suspicious of him, but the prosecution just wasn't able to present enough compelling evidence to convict him.
- have a compelling reason for (something)
Unless there's a compelling reason to stay, I say we call it a day and head home.