Engasgado literally means "choked" or "stuck in one's throat." It's a vivid word that goes beyond a simple "sad" or "upset."
In the song, the singer uses it to describe how the final goodbye, "aquele tchau," is still "engasgado aqui." This powerfully conveys the lingering pain and the inability to fully process or accept the breakup, as if the words of farewell are physically stuck and suffocating him.
"Despedida De Casal" (A Couple's Farewell) is a bittersweet Brazilian sertanejo song about a breakup dressed up as a nice night out.
A couple goes to a fancy restaurant, wine, ceviche, she's done her hair and dressed up, and she spends the evening smiling. But the elegant setting is a cover for what the dinner really is: the moment they end things. She walks away without shedding a single tear.
The singer isn't so composed. Even after it's over, he still hasn't come to terms with the ending, and he can't get the word "goodbye" out; it stays stuck in his throat.