You’re misparsing the sentence. It’s not “they threw me off the float” (i.e., ‘I’ was previously on a float with some other people, but then they threw ‘me’ off), but rather that ‘they’ (people on the float) threw ‘me’ something.
She’s talking about a parade going through town—that’s the Carnival mentioned a few lines further down. Carnival parades usually have people on floats, and often these people throw little tidbits into the crowd of people watching the parade: flyers, sweets, stickers, things like that.
Fay had clearly been outside where a parade was passing by, and someone going by on one of the floats had thrown something down to her from their position on the float. A few pages later, after an interlude in which Laurel recalls how Fay had come into her father’s life, we are told (somewhat indirectly) that the thing someone threw down to her from the carnival float was a green stiletto shoe.
(I haven't read the story, but Sarriesfan mentions in a comment that the story takes place in New Orleans in Louisiana in the south of the US. New Orleans is famous for it’s huge celebration of Mardi Gras which includes hundreds of such parade floats. The Carnival talked about here is likely a Mardi Gras celebration.)