When I try to install VersionPress, it tells me, “The package could not be installed. No valid plugins were found.”

It looks like the file you downloaded, versionpress-master.zip, isn’t structured like a WordPress plugin. When I unzip versionpress-master.zip, the plugin looks like it’s in the directory plugins/versionpress. You’ll need to copy that directory and all its contents to your site’s plugin directory (usually wp-content/plugins).

Edited to add: The official site’s installation instructions indicate that you should download the plugin zip file from their site, not from GitHub.

Edited again to add: The plugin is no longer being developed and does not support WordPress 5.x/Gutenberg.