Apple doesn't want you to use anything not coming from the AppStore or from developers who pay over $100 a year to Apple.
There are however some security setting you can set less agressive. The first time you run the app, click it with the right mouse button or hold ctrl-and-click if you have a one button mouse, and then select "open"... you will still get a warning, but the choice to open it anyway, and when you open it again later, the issue should no longer be there.
Apple is just a little paranoid, or money hungry... your choice.... If the itch.io app does bypass this, that would be good. For me this was a reason to quit Mac.