When Travis is still obsessed with winning Betsy over—after disturbing her by taking her to a porn flick on their first date, that is—he tries to make it up to her by buying her flowers. He doesn't know her address, though, and he ends up having a bunch of bouquets lying around his room. They start to rot and he says that their smell is giving him a headache. He also imagines that he has stomach cancer.
The decay of the flowers definitely mirrors Travis' own mental state of decay. Since he's failed to connect with another person—a person who would've accepted the flowers—they stay in his room (like he does so frequently) and fester. They're a good thing gone bad, which maybe implies that Travis really was, originally, a good person who is now rotting on the inside.
Finally, Travis gets to a point where he ritualistically burns the remaining flowers—indicating the he's put Betsy and the whole human dream of love and connection behind him. Now, he just wants to wreak vengeance on New York—kill some people.
Martin Scorsese says he intended to film this as being like Catholic Mass—with Travis lighting shoe polish on fire before burning the flowers, kind of like a priest lighting candles. However, this is a parody of Mass—instead of being a ceremony of life, it's a ceremony of death. (Scorsese says he was aiming for the same sense of a Mass-like ritual in the scene where the gun dealer lays out guns on a bed like he's preparing an altar).