Chingar a tu madre is a very strong and offensive Spanish idiom, roughly equivalent to "go f*** your mother" or "go to hell" in English. It's a highly vulgar expression of dismissal and contempt.
In "Felicidades," Grupo Firme uses this phrase to express extreme anger and rejection towards a former lover. The singer tells the person they've "won a raffle ticket to go chingar a tu madre," emphasizing the depth of their disdain and the finality of the breakup. While crude, its inclusion makes the song's emotional intensity undeniable and memorable.
This is a breakup song dressed up as sarcastic congratulations. Grupo Firme and Alejandro Fernandez trade verses mocking an ex-partner, thanking her for the "prize" of being free of the relationship and wishing her well with heavy irony.
The humor gets sharp and vulgar in places (the kind of blunt language common in Mexican regional music), which is part of the joke: the singer is not really happy, just performing confidence to save face after being hurt.
By the end, the tone softens a little into affection for the group and the genre itself, before circling back to the same mocking "congratulations" that opened the song.