F-Droid is free, to download it you just click the “Download F-Droid” button on https://f-droid.org/. This downloads a .apk file which you open to install F-Droid. How the opening exactly works depends on your browser, but normally you should get some kind of dialog that the download was completed with a button to open it.
After the download completes you get redirected to Donations | F-Droid - Free and Open Source Android App Repository, explaining that F-Droid is powered by donations, with instructions on how to support the project if you want.
I am not sure what you exactly did after that to accidentally donate $2.00 to F-Droid (which is distributed between members of the F-Droid team and by pure chance came my way), but that is what you ended up doing. I would suggest you contact Liberapay about this and explain the situation, also to make sure you haven’t accidentally set up a recurring donation (I can’t verify this for you): Contact - Liberapay.