It is sort of tricky to run spatial queries against tables in a GeoPackage anywhere, but especially outside of QGIS. This is because we're reliant on Spatialite to carry out those queries and Spatialite is just not well supported. A real shame!
Luckily there are a few resources for helping you get up and going with installing Spatialite for working with GeoPackages in a SQL client like DB Browser.
There is a page in the DB Browser Wik on GitHub called " SpatiaLite on Windows" and it is dedicated to installing Spatialite for use with DB Browser. Note that there are special instructions for a Windows 10 issue.
Bryan McBride at Spatial Networks wrote up a nice piece on working with GeoPackages in DB Browser called "Working with Geospatial Data: An Introduction". It's got a nice walk-thru of how to use DB Browser with GeoPackage as well as some general installation instructions.