I've never had a problem downloading and using the map data in OffMaps:

- position the map where you want to download the maps for
- tap on the centre button on the toolbar
- select download map only
- drag out an area
- adjust the slider for detail required
- tap on start download
- wait
- tap on close when complete

That's, I now have the data cached on the phone (as proved by putting it into airplane mode and killing the app/restarting it).