Why do Orthodox Jews wear felt hats?

Orthodox Jews cover their heads as a reminder that God is above. This can be done with a kippah or yarmulka (cap). Wearing a hat is a sign of dignity, just as you wouldn't be surprised to see a senator wearing a hat.

Why some Orthodox Jews wear a big hat, is perhaps based upon the European region they originated from (Poland, Galicia, etc.), and/or to be recognizable as belonging to a particular subgroup within the various Hassidic customs.