On Android, files are stored in the MapsWithMe folder in your device memory or on SD card in Android/data/com.mapswithme.maps.pro/files/MapsWithMe folder, if you store downloaded maps on SD and have Android 4.4 or newer.
Maps are stored in a subfolder with map data version in its name, for example, 170817. Bookmarks are stored in a root of MapsWithMe folder and have .kml extension, for example, My places.kml. You can access the folders with the help of almost any file manager like Samsung file manager or Total Commander.
On iOS, you can access the app files with iTunes: connect your device to a computer → open iTunes → select your device → File Sharing → select maps.me.