By appealing to their hatred for the British and economics.

First economics. Benjamin Franklin had made several promises to France to give them priority when it came to trade. This was massive as most of the powerful nations of European were receiving their raw goods from the Americas, and this was especially true of Great Britain and France. Great Britain was the largest naval power in the world, and it ws almost entirely financed by Britain ability to absorb natural resources and ship out products (often right back to the Americans

By taking trade away form Great Britain, France could: 1) Receive substantial more natural resources then before 2) Reduce the economic strength of the British 3) Reduce Britain's access to natural resources (much of Britain's ships were made with American lumber)

The second point I want to bring up is France's hatred for England. To make a generalization, France and England had been warring with each other on and off for centuries. In most of these conflicts, they were hegemony power that dragged in various allies.

Now France at this particular time (American Revolution) could not truly contend with England. They held the strongest Navy ever, they held the strongest economy ever, they held the most land out of any empire (before or since), they had the largest army in the world. So in short, they were not to be trifled with directly. What better way to weaken your enemy then have someone else fight for you?

TL;DR: The enemy of my enemy is my friend, and France wanted to weaken the British economy while strengthening their own with trade rights