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 bookmarks subfolder and have .kmb extension*, for example, My places.kmb. You can access the folders with the help of almost any file manager like Samsung file manager or Total Commander.
*Note: to get bookmarks in KMZ (KMZ is a zipped KML) format you should export them directly via the app. 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.