You're really asking two questions.
The links gleek posted explain what the term 'FPGA' is, as a technology. It is the technology used to create custom-programmed chips, which reproduce the circuits of the original hardware PCB's for whatever game you're interested in, but in a much smaller form, where a whole 1980's-era PCB can be shrunk into a single modern FPGA chip. FPGA technology can be used to create many different types of chips, for many different applications, in many industries. They are basically generic pieces of programmable hardware.
However here in arcade hobby land, we use the term 'FPGA' slightly differently, where it refers to a multi-board that someone created, using FPGA technology.
In terms of using an FPGA-based arcade board in a scratch-build cab, you would use it just like any other arcade PCB. Most of the FPGA multiboards are JAMMA, so you treat them just like any other JAMMA PCB, and you need a JAMMA power supply, harness, and control setup, just like any other JAMMA cab.
FPGA is just the technology that was used to create the board (i.e., to reproduce the circuitry for the game board being replicated, in a single chip or a small set of chips, which are created by the designer by writing FPGA code, which defines the digital circuits in the chip). But you don't need to know or care anything about that in order to use an FPGA-based arcade multi-board in a cabinet. (Just like you don't need to know anything about being a chef or nutrition, to go out and eat in a restaurant.)
Additional details will also depend on which FPGA board you're looking to use. Boards like the Bitkit and Jrok multi-boards come as prepackaged products, that are basically plug and play JAMMA boards. Other FPGA-based boards like the Mister, require more manual configuration and setup. But ultimately they all just plug into a harness, and connect to a PS, CP, and monitor, like any other arcade board.