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.
Notice: 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.
Enjoy!
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