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?
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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:
1932 W. R. BURNETT Silver Eagle i. 7 ‘He don't even pack a heater.’ ‘Don't what?’ ‘He don't carry a gun.’
And the first citation I see that uses heat instead of heater (from an old dictionary):
1942 L. V. BERREY & M. VAN DEN BARK Amer. Thes. Slang §496/6 Pack (a) heat,..to carry a gun.
EtymOnline offers this:
Sense of "to carry or convey in a pack" (1805) led to general sense of "to carry in any manner;" hence to pack heat "carry a gun," underworld slang from 1940s.
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:
Since reading Little Caesar we chat lightly with gangmen about packing a heater (carrying a gun), or getting the book for a caper (a life sentence for a murder). . . . The surprising thing about this book of WR Burnett's is that there are three hundred-odd pages of this slang set in prose as staccato as the frequent machine-gun fine, and yet it manages to be an excellent novel.
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:
And he never packed a heater, as do many Chicago police reporters. He was abstemious. A glass of beer was his limit. Gambling was a consuming fever with him. It debilitated his moral being. It enslaved him like a drug.
Scribner's magazine: Volume 87, 1930:
Felix stood looking from Mary to Oregon, fascinated, unable to move. “Fast rambler, yeah,” said Mary. “Tough guy. Picks on old 'boes that can't lift a hand and him packing a heater. Yeah, tough guy.
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:
”Yeah, but it’s just a stall. He don't even pack a heater."
"He don't carry a gun. It's all a bluff. It's the old run-around."