Since the update to Google Maps last year you should now use Google Maps Engine to create custom maps and access these on Android via the corresponding Maps Engine app.
From Desktop Maps you can get to Maps Engine by clicking in the search box and then clicking 'My Custom Maps'.
To import existing maps saved to 'Classic' My Maps:
- go to 'My Custom Maps'
- then 'See all my custom maps'
- in the 'Open a map' window, select 'Classic My Maps'
- tick the maps you want to import and then 'Select'
- followed by 'Import'
You should then be able to find these maps in the Android Maps Engine app under 'Open a map' > 'Classic maps'.
If your map wasn't created by you (will show as 'bookmarked' rather than 'created by you' in My Places) then you might need to re-import it into 'My Places' as KML to make it available to import into Maps Engine...
- Click the name of the map in 'My Places' (found by clicking the little cog icon in the bottom right of 'new' Maps or at https://www.google.co.uk/maps/mm)
- In the left hand panel copy the link to the KML file
- Back at 'My Places', click 'Create with classic My Maps'
- Give the map a title then 'Import'
- Paste in the URL of the KML link and 'Upload from URL'
You should now be able to go back to Maps Engine and import the map as above.