From the docs

Console / text-mode applications

By default, the itch app redirects the standard output and standard error to a log file on disk, which helps debugging when reports are sent.

I want to tail that log. Where can it be found?

In itch v25, the main log is in $XDG_CONFIG/.config/itch/logs/itch.txt on Linux, %APPDATA%/itch/logs/itch.txt on Windows, ~/Library/Application Support/itch/logs/itch.txt on macOS.