For a hundred francs a year, she did the cooking and the housework, she sewed, washed and ironed, she could bridle a horse, fatten the poultry and churn the butter, and she was unfailingly loyal to her mistress, even though the latter was not a pleasant person. (1.2)
Humility is a funny word. It can mean humble, but also has a little to do with being humbled, as in humiliated. We're not saying that Madame Aubain was constantly humiliating Félicité, but Félicité was always in an inferior position to Madame Aubain and had a very submissive attitude toward the boss-lady.
By the time she reached her fifties, she was ageless—and, with her constant silence, her upright figure and her measured gestures, she seemed like a woman made out of wood, a kind of automaton that moved without thinking. (1.8)
If the opposite of humility is pride, then Félicité definitely falls on the humble side of that spectrum. Her 'agelessness' shows that she doesn't worry about her appearance; she's all business. And her mechanical movements are just the result of years of work—it's not like she's showing off her mad housecleaning skills or anything.
[. . . H]e proposed marriage. She did not believe him at first, but he swore solemnly that he meant it. (2.9)
When Théodore proposes marriage to Félicité, she doesn't believe him. Not because he's one to joke, but because she can't imagine someone like him would want to marry someone like her. It's her humility—that sense that she is beneath everyone—that keeps her from imagining anything good for herself.
His white cravat, his bald head, the jabot of his shirt, his roomy brown frock coat, the way he bent his arm when he took snuff – everything about him produced in her the kind of agitation we feel in the presence of extraordinary men. (2.23)
Monsieur Bourais is an ex-lawyer, and…well, you know the shtick: lawyers can be total know-it-alls. In fact, we'd recommend that most of them eat some humble pie themselves. So it's no surprise that simple Félicité feels agitated around Bourais; his presence alone is enough to remind her that she's inferior to him.
Paul explained these engravings to Félicité. It was the only book learning she had ever received. (2.26)
A big contributing factor to Félicité's humility is her lack of education. Whether religious or academic, she grows up without anyone worrying about what she learns and how. This also means she hasn't really been taught how to act in society. That in and of itself keeps her pretty humble, we bet.
For many years, this episode was a topic of conversation in Pont-l'Évêque. But Félicité, unaware that she had done anything heroic, took no pride in it. (2.37)
Félicité protects the Aubain family from a charging bull like it ain't no thing. The whole family could have died, especially her, but she "takes no pride" in her own bravery. She could have become a blowhard, bragging about her courage for the rest of her life, but her humble self forgets it.
After making a genuflexion at the door, she would advance along the high nave between the two lines of seats, open Madame Aubain's pew, sit down, and look around her. (3.1)
It's interesting the way that humility is just sort of built into the fabric of Félicité's everyday life. She's the servant in Madame Aubain's house, so she's clearly below everyone there. Then, when she goes to church, she makes a genuflexion (kneels) to show her humility before God.
As for the dogmas, she did not understand a word, and did not even try to understand. (3.5)
Dogma is religious teaching made up of different statements of what the Catholic Church believes. The fact that Félicité does "not even try to understand" them indicates that she is aware of her own ignorance; she knows that she's below understanding what so many priests have learned.
Bourais threw up his arms, sneezed, and laughed uproariously, his mirth aroused by such innocence. Not that Félicité understood the reason why: her intelligence was so limited, she might have been expecting to see a picture of her nephew on the map! (3.44)
When Félicité finds out that her nephew is in Havana, Cuba, she worries about him and wonders whether he might be able to get home by land. When she goes to Bourais for help, he laughs at her, and she doesn't even know why. Félicité's humility comes from her lack of experience and knowledge of the world. And just in case you're in the dark on this one: You'd definitely have to cross an ocean if you wanted to get from Cuba to France.
Félicité was as grateful as if her mistress had done her a good deed, and from this point on cherished her with an animal-like devotion, a religious veneration. (3.90)
There it is. We've been tiptoeing around the idea, but the truth is that Félicité doesn't act like a human being. She's below everyone else, "animal-like." She treats Madame with "religious veneration," meaning she treats her like a god. If humility means lowering yourself in the face of others, then Félicité is the queen of it.