Even among the ancients the container often signified the thing contained; the Romans used olla, pot, many times instead of the meat within the pot, and so did our own forbears.
Hence, when they said that they must keep the pot boiling, they meant that they must have something within the pot, which, when removed, would be edible; that is, that they must supply meat or other material for a stew, provide a livelihood.
This was the only figurative meaning from the sixteenth to the nineteenth century; it gave rise to such allied sayings as “to go to pot” (to cut up and prepare for the pot; hence, in present usage, to become disintegrated), “potboiling,” (doing something, usually something of no great merit, that will provide for one’s immediate needs).