The lion has been associated to the city of Lyon, France since the Counts of Lyon adopted the animal in their blazon in the tenth century.

The main sponsor of the Tour de France is LCL or Le Crédit Lyonnais, a financial institution originating from Lyon, so the lion is there to remind us of the sponsorship in a more subtle and discrete fashion than the flashy LCL caravan in the Tour de France.

Note that the name of the city of Lyon comes from its former name "Lugdunum", which later changed to "Lyduum" and bears no relationship with the "lion" animal, even though the lion present in the city's coat of arms might have influenced the francization of the name over time.
The company that sponsors the tour "credit lyonnais" gives them to the winner of the stage. The lion is to represent "Lyonnais"