PS1 emulation on the PSP is actually mostly a native Sony feature! Sony has always been very interested in backwards compatibility; initially to solve the chicken-and-egg problem of a new console not having many titles available at launch, but later largely focusing on additional revenue by selling people digital downloads for games they might well already own (because they removed native media compatibility...)
For the PSP, media compatibility was obviously never a possibility, but as I understand it, the native PS1 emulator on the PSP essentially works with completely unmodified PS1 ISOs in some sort of container file format, which homebrew developers have then quickly reverse engineered.
The emulator itself is technically quite impressive, and apparently largely runs unmodified PS1 MIPS R3000 code on the PSP's MIPS R4000, although the GPU is emulated [1].
For the PSP, media compatibility was obviously never a possibility, but as I understand it, the native PS1 emulator on the PSP essentially works with completely unmodified PS1 ISOs in some sort of container file format, which homebrew developers have then quickly reverse engineered.
The emulator itself is technically quite impressive, and apparently largely runs unmodified PS1 MIPS R3000 code on the PSP's MIPS R4000, although the GPU is emulated [1].
[1] https://www.eurogamer.net/investigating-the-psps-psone-emula...