Ensorcelle comes from the verb ensorceler, meaning 'to bewitch', 'to enchant', or 'to cast a spell on'. It's a beautifully poetic word you won't find in your average textbook.
In the song, the Seine river is personified as a captivating woman. The singer says, "Tellement jolie elle m'ensorcelle" (So pretty she bewitches me), perfectly capturing the magical, enchanting feeling of falling under the spell of Paris and its famous river.
Vanessa Paradis and Matthieu Chedid sing this duet for the animated film Un monstre à Paris, and the whole song is a crush on a river. The Seine wakes up like a woman who knows she looks good: she "sort de son lit" (gets out of her bed), which in French is also what a river does when it floods. She is "Tellement sûre d'elle" (So sure of herself), and "Tellement jolie elle m'ensorcelle" (So pretty she bewitches me). Then comes the best joke in the song: "Tu n'es pas saoul" (You're not drunk), "Paris est sous" (Paris is under) the Seine. You might think the city looks upside down because you've had a few, but no, that's just Paris reflected in the water.
Listen for the pun in the middle verse. "La Seine" (the Seine) and "La scène" (the stage) sound exactly alike in French, so the river turns into a stage for a performer, "Extravagante quand l'ange est sur" (Extravagant when the angel is on) it. The chorus shrugs off any explanation: "Je ne sais, ne sais" (I don't know, don't know) "Ne sais pas pourquoi" (Don't know why), "On s'aime comme ça la Seine et moi" (That's how we love each other, the Seine and me). The ending stands on a real Paris footbridge, "Sur le Pont des Arts" (On the Pont des Arts), where "Mon coeur vacille" (My heart wobbles) "Entre deux eaux" (Between two waters), a French idiom for being caught halfway, not sure which way to go.