Database Inspector

It seems some improvements have been made in Android Studio since this question was asked and the previous answers were provided. There is now a built in way to look at the database while debugging your app without needing to download files or search through the device manager.

How to Access Database Inspector

View > Tool Windows > App Inspection. If you haven't already, start debug mode on your app. Choose the device and app being debugged from the drop down list. Wait a few moments for the view to refresh. Select the tab for Database Inspector. Under your database you will see the different tables in your database. Double click on a table to open it for viewing and editing individual existing values.