You cannot download music directly from the internet onto a PSP using built-in features. The PSP's browser and network capabilities are too limited for modern streaming or direct downloads.
Why Direct Download Doesn't Work
No built-in music downloader, streaming app, or modern codec support.
Most legal downloads have DRM that prevents playback on unauthorized devices.
The PSP's SSL/TLS support is outdated; many sites won't load.
Workarounds (The Practical Way)
Transfer from Computer (Recommended):
Download music legally (Amazon Music, iTunes, Bandcamp, YouTube Music) to your PC → Connect PSP via USB → Copy MP3s to MUSIC folder on Memory Stick.
Homebrew Apps (CFW Required): Custom firmware enables apps like Music.prx plugins or PSP Radio for Wi-Fi streaming. Research compatibility for your firmware version.
Memory Stick Direct: Load MP3s onto a Memory Stick Duo via card reader → Insert into PSP.
Notes
Respect copyright laws; ensure files are MP3 format (PSP does not support AAC, FLAC, OGG natively).
For modern convenience, a smartphone/tablet with streaming apps is vastly superior.