So are you saying I don't need to buy the specific development board for this module, the ISP1507-AX-TB?
ISP1507-AX-TB is not a module. It is a development board with a module soldered on it.
I can use any board with J-Link?
J-Link exists in two versions - as a standalone programmer in a plastic box, and as an IC that dev kit vendors can solder on the board.
On the website where they sell the dev boards for this module they say "with integrated J-Link OB JTAG/SWD Emulator". What does that mean? Is that what I'm looking for?
This is the later. You can buy that board, which comes as part of the ISP1507-AX-EB. You can use that to program the devices.
But as I said, you are wasting your time. If you can't figure out what is module and what is the kit, there is zero chance you will figure out how to program it. Hardware is by fat the easiest part. You need to start with basics and build your way up to the more complex systems.