Here are step-by-step instructions (mostly taken from a combination of the other answers). This works even on devices that are not rooted.
Connect your device and launch the application in debug mode.
You may want to use
adb -d shell "run-as com.yourpackge.name ls /data/data/com.yourpackge.name/databases/"to see what the database filename is.
com.yourpackge.name is your application package name. You can get it from the manifest file.
Copy the database file from your application folder to your SD card.
adb -d shell "run-as com.yourpackge.name cat /data/data/com.yourpackge.name/databases/filename.sqlite > /sdcard/filename.sqlite"
Notice: filename.sqlite is your database name you used when you created the database
Pull the database files to your machine:
adb pull /sdcard/filename.sqlite
This will copy the database from the SD card to the place where your ADB exist.
Install Firefox SQLite Manager: https://addons.mozilla.org/en-US/firefox/addon/sqlite-manager/
Open Firefox SQLite Manager (Tools->SQLite Manager) and open your database file from step 3 above.