First of all, it seems unlikely that running that script is going to be the right thing to do but I havenât personally reviewed it so I canât say for sure.
What I can say for sure is that the command you ran doesnât install anything. It changes the permissions on the file to make it so you can run it.
If you are sure that you want to run it you can do that now by switching to the directory containing the file and typing ./displaylink-driver-5.5.0-59.151.run
That means the permissions were successfully changed.
EDIT: I guess one more thing I should add is that, generally speaking, you donât install software on an Arch-based distro by downloading files and installing them manually. You can usually find everything you need in either the repos or AUR. That is generally where software and drivers come from.
displaylink is a package in the AUR--that might be what you need in this case.
AUR packages have to be built; I recommend using the AUR helper paru, which is included with Garuda Linux by default.
paru displaylink
It will present you with a list of available packages and you can select the one you need, then paru will help build it for you.
You may need to refer to this ArchWiki article about setting up a DisplayLink device, because in this case I'm afraid there is a bit more to it than just installing the correct driver: DisplayLink - ArchWiki
We encourage a kind of netiquette aimed to marking as Solution the post which provided the needed information, the right way to go... Not the final implementation.
If the one I marked is not correct, you can still mark another post or provide the final solution.