Foudroie is a powerful verb that literally means "to strike down with lightning", from the noun foudre (lightning).
In the song, the line "C'est ça l'amour, ça te foudroie" (That's love, it strikes you down) uses this word to describe the sudden, intense, and all-consuming feeling of falling in love. It's directly related to the famous French expression un coup de foudre, which is how you say "love at first sight".
Picture this: it is nighttime in the French capital, the city lights turn on, and the streets empty out. Instead of heading home, Monroe stays right there on the pavement. In "Regarde", she wanders through Paris searching for a lover across every neighborhood, singing every single day. She uses the word "faubourgs" in the line "J'ai cherché ton amour / Dans tous les faubourgs" ["I looked for your love / In all the outer districts"]. Historically, a faubourg was a suburb outside the city walls, but today it names specific old neighborhoods like Faubourg Saint-Honoré. It gives her search a classic, sprawling street feel.
Her chorus hits you directly in the face. She sings, "C'est ça l'amour / Ça te foudroie" ["That is love / It strikes you like lightning"]. The verb foudroyer comes from la foudre (lightning). French speakers use it when love hits someone suddenly and violently, the way English speakers talk about being thunderstruck. Then she drags her love straight out "sous la pluie battante" ["under the pouring rain"], swearing to tell every man, woman, and child that love is everywhere. It is big, dramatic popera that refuses to be quiet about devotion.