Because you cannot use the Google Play Store (it will either show "incompatible" or a broken old version), you must sideload the APK. Follow these steps carefully.
Have you successfully run Messenger on Android 2.3? Share your version number and device model in the comments below (if your browser still loads Disqus). facebook messenger old version apk for android 2.3
This error indicates that the APK file requires a higher version of Android than 2.3. You must delete the file and find an even older version of the app. Connection and Login Failures Because you cannot use the Google Play Store