A It is variously said to come from the habit of beggars of soliciting contributions by thrusting out tin pans, into which generous passers-by would place their coins; or perhaps it's from the Spanish pan, literally meaning “bread” but which could also mean “money” (much as our word bread can in modern English), or ...

How did the term panhandler come about?

Meaning "an act of begging" is attested from 1849, perhaps from notion of an arm stuck out like a panhandle, or of one who handles a (beggar's) pan. panhandle (v.) "to beg," 1888, from panhandle (n.) in the begging sense.

Is the term panhandler offensive?

The term is fairly derogatory, but it's commonly used for those whose mainly support themselves this way. It's best to avoid vaguely offensive terms like panhandler and beggar altogether — people asking for handouts have identities beyond begging for money or food.