First, I suppose you are referring to one of these two representations of Louis Gillet : Le Brave Louis Gillet or Le Vaillant Louis Gillet. A couple of other representations exist.

At the time of Gillet's adventure with these two bandits, the Maréchal des Logis ( if you're American, think of someone like a Staff Sergeant ) was 77 years old. The story is simple, the old soldier defeats the two bandits on his own and brings back the lady to her parents. Her father offers that both of them get married, to what Gillet says that he is too old to make her happy. So, the father decides to give Gillet a purse, but Gillet answers : "the reward is in my heart".

You have everything here to make a beautiful story with a virtuous, even chivalrous, old soldier.

Now if you look at this representation you will see more interesting information. It's the same artist, Borel, you went to Gillet's house and made a portrait of him but here it was engraved by Etienne Claude Voysard, a Parisian engraver who worked for ( as said ) Monseigneur le comte d'Artois, Frère du Roy. The count of Artois is the future Charles X, brother of Louis XVI and the commander of the Gillet's Regiment ( the Regiment of Artois ). With that said, you have an explanation for having so many representations of Gillet's story. It must have been pleasing to have such a soldier in your regiment blablabla ...

But you can see beyond that. It happened in the late 1780s. A time where the notions of Virtue and Reason ( the Enlightenment ) were important. Louis Gillet is an old man ( age is wisdom ), from a very simple/humble background, treated in the Royal Hospital of the Invalides ( an hospital for old soldiers too poor to live by themselves ). The person of Gillet represents a perfect hero. But he was not the only one in this case.

You may have heard of Jean Thurel whom Napoléon described as the oldest soldier in Europe. Thurel died in 1807 aged 108, after a career of no more than 90 years. His story impressed a lot of people, even before the Revolution and he had been invited by Louis XVI to his palace. The King called him father and offered Thurel a lot of rewards including a domestic to tend the old soldier's care, which Thurel humbly refused. There are other examples.

To conclude, Gillet's story fits in the kind of stories people liked back then. Virtue, old men's wisdom, chivalrous behaviour. In 1791, deputy Nicolas-Louis d'Estagniol, who had been Gillet's superior officer in the Regiment of Artois proposed to the Assembly that a pension should be given to him. But think of it as the very good story people liked. Just like when today a police officer or fire fighter runs into a burning building and save a kid and a puppy. It makes the headlines.

Source : it's hard to source my answer. Most of the information were on the images ( links from Gallica in the answer ). I also used the French National Assembly's website. For Thurel's story, the Léonore database listing Thurel's Legion of Honour papers.