Franchise is a word that became popular with hamburger joints, such as McDonald's, Burger King,White Castle, etc. in the middle of the 20th century. Basically, it means a business which has the logo and product of a big company but is individually owned. The owner has his store designed and built by the big company according to their specifications, (that's why they all look alike), and sells the company's products, but he keeps the profit minus whatever set amount he pays the company for the use of their name, etc.
Since that time, franchise has grown to mean almost any arrangement where the product is made by one big company but the store or distributorship is individually owned. Not sure how this applies to video games, unless someone developed a video game using another game's characters or settings and is paying the people who own those characters money to use them in the new game.