Of course we don’t cut the beaks off, but we do “trim” the tips of their beaks. This is for a similar reason that we humans trim our finger and toenails: if we don’t, we can injure ourselves, or those around us. Cats and dogs have their nails trimmed for the same reason.
Does it hurt the animals, in our case the birds, when we trim their beaks? It probably does a bit, in a similar manner to when you trim your fingernails too close. This is typically now done at the hatcheries with a laser beam before we ever buy our birds. If we have to do it on the farm, as a preventative measure, the birds are fed extra vitamin E and K to enhance coagulation (to prevent bleeding), if bleeding were to occur.
You might be interested to know that chickens raised for commercial production that do not have their beaks trimmed exhibit a very troublesome characteristic: they become cannibalistic. Their inherent “pecking order” comes out with the strongest ones picking on, and often causing very serious injury to others. This is particularly true in non-caged systems where more birds co-mingle with each other.