Usually to get access to beta tests with Android applications you can sign up via the Play Store. However the new version of eBay 188.8.131.52 is a closed beta, and you need to be authorised when opening the application. So simply just installing the APK wont gain you access.
However there is a simple form to fill out via Google Docs. All you need to do is input your email, preference (Android or iOS), and country. You should receive an email with a download link when there is a new test build.
The new layout of eBay for Android is amazing, and is definitely a welcome upgrade!