I figured this out already. In case anyone's having the same problem, the solution is to use 'cdvfile://localhost/persistent/' to get to the root folder in Phonegap. So, in my case, the change would be,
var fileTransfer = new FileTransfer();
fileTransfer.download(encodeURI('http://services.arcgisonline.com/ArcGIS/rest/services/World_Street_Map/MapServer/tile/18/96730/70400'), 'cdvfile://localhost/persistent/com.mee.flight/maptile/70400', function (entry) {
//Do something
});