What kind of breadboard and prototyping board do I need for this right-angle DB25 connector?

What you are searching for is a DB25 breadboard adapter, looking like this:

Website (example): Example

They also exist in a smaller (double row) format like on this website

That type of staggered pins are more or less standard because they must also fit straight-through DB-25 footprints which are standardized, but you need a protoboard that is designed to hold them. This image was the first I found on google, but you can find them in all shapes and forms. This is for DB-9. Note the odd staggered part to the right:

Since the normal protoboard spacing is 2.54 mm while the DB-25 pin spacing is 2.77 mm, you need to search for a dedicated DB-25 protoboard. It could be easier to buy an adapter board which includes the correct connector, but have pins or screw terminals at a more convenient spacing.