What does the phrase "and ignobly garnered largesse of great cities" mean in "Compliments of the Season?"

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.