I believe it means “to carry a weapon”, but I would also like the phrase origins, if possible. So the full question is:
What is the meaning of the phrase “packing heat” and what are its origins?
Heat/heater is slang for "gun" (definition #14) and to pack has an informal meaning "to carry, deliver, or have available for action" (v. tr. definition #8). So "packing heat" means that you are carrying a gun (and are ready to use it).
According to the Oxford English Dictionary, here is the first citation for this phrase:
And the first citation I see that uses heat instead of heater (from an old dictionary):
EtymOnline offers this:
There's a few examples in Google Books of "pack a heater" earlier than the OED's 1932.
The Golden book magazine: Volume 10, No. 57, September 1929:
Burnett "is best known for the crime novel Little Caesar , whose film adaptation  is considered the first of the classic American gangster movies." The OED 1932 is from Burnett's Silver Eagle.
Al Capone: biography of a self-made man by Fred D. Pasley, 1930:
Scribner's magazine: Volume 87, 1930:
And in fact the OED's Burnett citation appears to have also been printed in a magazine a year before, in Collier's: Volume 87, 1931:
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