Let's look at the individual words.
- "Ignoble" means immoral or in some way bad.
- "Garner" means to acquire or get.
- "Largesse" means something like "riches" -- things that the rich have that they can then give away.
So if you put those three together, you can see what O. Henry is saying. Fuzzy is a beggar and one of the things he subsists on is the "ignobly garnered largesse of great cities." In other words, part of what supports him is handouts from rich people (largesse) that was gotten in immoral ways.
By saying this, O. Henry is arguing that the rich of this era (the "Gilded Age") were "robber barons" who got their money by exploiting others. So this is a bit of social criticism that is put in to this amusing story.