Well, I'll start with the DS homebrew question first.
First, you'll need, as it says in the article for the DS pixel fixer "You need a passme, flashme or wireless multiboot available to run it on your NintendoDS." Which is basically, an R4DS, or an M3 (theres various M3 cards but I think the newest/best one, is M3 Lite)
Once you've got one of them, you're also need a mem card - micro SD I believe for those 2, but it depends which one you purchase.
Now, I haven't actually run any homebrew on my DS, but I do believe it is as simple as putting the homebrew folder on the mem card. Either way, the R4DS or similar is needed first.
As for
PSP homebrew, every
PSP firmware version can be downgraded to 1.50. From 1.50 you can run homebrew and isos. But for isos you need an extra homebrew app and doesn't run some of the newer games, ones that require higher firmwares. So, what most people do now, is upgrade to custom firmware. Which is basically, all the features of the official sony firmware, with extras, such as allowing you to run homebrew, isos, and PSX games. The list you can run on custom firmware is far greater than the official list.
Idk how much you know about what homebrew is, but, its basically, an app made at home, not official. It can range from useful apps like the pixel fixer, to custom made games, to emulators. With regards to the
PSP, you can emulate Nes, Snes, Megadrive, Master System, PS1, GBA, N64 and others. Some work better than others though of course.