Best way - Payload Manager on the console. Load pldmgr.elf from the R2 panel once, then in its dashboard add this as a custom repository:
http://138.199.213.155/ps5-repo.json
It will list every payload below with its description, install them on the console, and can auto-load whichever ones you pick every time it starts.

Or, to load one right now, use the console's own launcher.
Load websrv once from the R2 panel in the exploit page, then open:
http://127.0.0.1:8080/elfldr.html  (from the console)
http://<console-ip>:8080/elfldr.html  (from a PC or phone on the same network)
Drag a payload onto that page (or pick one) and press Send. That works any time, survives reloads, and needs no exploit re-run. websrv also gives you a homebrew launcher (/index.html) and a filesystem browser (/fs/).

Why the downloads below do not install themselves: a browser saving an .elf just saves it. Payload delivery has to be pushed at the console - either through the launcher above, or through the R2 panel on the exploit page (which only exists on that page, because it writes through the kernel primitive), or from the console's own storage.

Option 2 - no browser at all: download payloads.zip on a PC, put the ELFs on a USB stick or FTP them on with ftpsrv (R2 panel, port 2121), then load them from etaHEN's toolbox → Plugin / Payload ELF Menu (it can auto-start too).
Option 3 - the R2 panel: stay on the exploit page and press R2; it reopens as often as you like while that page is open.
After a reboot nothing is running: open the guide, press RELAPSE, then R2 for the panel.