Installed with curl
If you installed Loom with the curl line, loom upgrade downloads the latest release for your machine from GitHub. It checks the download against the release's list of checksums, then swaps it in for the file you ran. A download that doesn't match is thrown away, and your Loom stays as it was. If Loom sits in a folder you can't write to, run the install line again instead.
Installed with a package manager
A package manager that installed Loom owns its files, so loom upgrade leaves them alone. It prints the command that upgrades Loom with that package manager instead, such as npm install -g @athrio/loom@latest. If you installed Loom into a project, the command it prints runs in that project's folder, with the package manager the project uses.
After upgrading
A new Loom expects each project's language servers to match it. Run in each project to bring them along.
If loom --version still shows the old version, another loom comes first on your PATH. Run to find it, and the command that removes it.