Background

Shell completion solutions have different ways of being loaded.
zsh specifically have elaborate competing plugin managers that deal with loading (or lazy loading) of shell extensions, as well as completion packages. (which I will not cover here)

What is common to all of these solutions, with regard to completion, is the loading of helper functions in the shell session (that's the code you saw being generated)

Each shell has a documented startup process. zsh starts by sourcing $HOME/.zshrc.

.zshrc will contain the loading code for every script session, this might include a plugin manager (which might load the completion code), but it can also include the completion logic directly.

The simplest way (although not necessarily the most effective one) would be to source (meaning loading the script's code into the session, as opposed to running the script).

You can (A) dump the output to a file and source that file, or (B) generate the source on the fly while the shell is initializing. (A) will be more performant, while (B) will be more dynamic (in case zellij updated, it will automatically load the 'new code').

But what does this all MEAN ?!

If you got with (A), you will first dump the completion into a script file:

$ zellij setup --generate-completion zsh > ~/zellij-completion.src

And then in ~/.zshrc, at the end of the script, have the following code:

. $HOME/zellij-completion.src

if you go with option (B), at the end of ~/.zshrc add the following code:

. <(zellij setup --generate-completion zsh)