To install the English version, download the *-English.dmg file (Hengband-22.214.171.124g-English.dmg is the latest) corresponding to the revision you want. Double click on the downloaded file to open the disk image, and then drag the hengband application to where you want to store it. Double clicking on the application will run it. Since the application has not been signed, you'll likely have to make an exception for it to use the application. Follow https://support.apple.com/guide/mac-help/open-a-mac-app-from-an-unidentified-developer-mh40616/mac
for Apple's instructions on how to do that. In short, one way to grant an exception is
To install the Japanese version, the installation instructions are essentially the same, but you'll need to download the *-Japanese.dmg file (Hengband-126.96.36.199g-Japanese.dmg is the latest) corresponding to the revision you want.