While I don’t have a Mac, that sounds to me like your Mac is configured to only run software if the developer has paid Apple for the privilege. Since most software available on itch.io is available for pay-what-you-want I would not imagine many developers have that kind of money to spare…
With that out of the way: I would imagine that there is a setting somewhere in your Operating System (probably Mac OS or Windows) that controls if software from “unverified” developers can be run/played. (There will be plenty of illustrated step-by-step guides online.)
However(!): Before you change that setting make sure you have antivirus software installed and running. Because changing that setting will allow you to run every piece of software that you can get on your computer - and you can’t always be sure if those programs are what it says on the label.