F = Field
P = Programmable
G = Gate <---- important
A = Array
You aren't "programming", you're "wiring gates together". In other words, you can build custom hardware to solve a problem without using a generic CPU (or GPU) to do it. FPGAs are implemented as a fabric of LUTs (Look-up Tables) which take 4- or 6- (or more) inputs and produce an output. That allows Boolean algebra functions to be processed. The tools you use (Vivado / ISE / YoSys / etc.) take a your intended design, written in a HDL (Hardware Design Language) such as Verilog or VHDL, and turn it into a configuration file which is injected into the FPGA, causing it to be configured to into the hardware you want (if you've done it right). FPGAs are a stepping stone between generic hardware such as a CPU or GPU and a custom ASIC. They win when you can express the problem in specialized hardware much better than writing code to do something on a CPU/GPU. Parallelization is the key to many FPGA designs. Also, you don't have to spend >$1MM on a mask set to go have an ASIC fabricated by TSMC, etc.
Given the density of the PDF, I saw AMD and AI in the title and assumed the scientific community was trying to get AMD GPUs to work. This makes more sense.
