Xamarin.Forms Saving file in filesystem

You are facing a permissions issue.

First, you will have to add in your AndroidManifest:

<uses-permission android:name="android.permission.READ_EXTERNAL_STORAGE" />
<uses-permission android:name="android.permission.WRITE_EXTERNAL_STORAGE" />

And since Android Marshmallow, you need to ask the user for the permissions, so I advise to use the package Permissions.Plugin

And add in your MainActivity:

public override void OnRequestPermissionsResult(int requestCode, string[] permissions, [GeneratedEnum] Android.Content.PM.Permission[] grantResults)
{
    PermissionsImplementation.Current.OnRequestPermissionsResult(requestCode, permissions, grantResults);
    base.OnRequestPermissionsResult(requestCode, permissions, grantResults);
}

You can check in runtime if you have the permissions by:

var status = await CrossPermissions.Current.CheckPermissionStatusAsync(Permission.Storage);
    if (status != PermissionStatus.Granted)
    {
        if(await CrossPermissions.Current.ShouldShowRequestPermissionRationaleAsync(Permission.Storage))
        {
            await DisplayAlert("Need storage, "Request storage permission", "OK");
        }

        var results = await CrossPermissions.Current.RequestPermissionsAsync(Permission.Storage);
        //Best practice to always check that the key exists
        if(results.ContainsKey(Permission.Storage))
            status = results[Permission.Storage];
    }

For further information you can check this blog post explaining all the permissions in Android - https://devblogs.microsoft.com/xamarin/requesting-runtime-permissions-in-android-marshmallow/

In addition to Bruno Caceiro's accepted answer, the UWP permissions you want to declare are as follows: DocumentsLibrary for documents, MusicLibrary for the music folder and so on.

You can get to these graphically via Right click project -> Properties and then clicking Package Manifest and going to the capabilities tab as shown in the screenshot, or alternatively right-clicking package.appxmanifest and selecting view code to edit the xml. See the code excerpt for how to declare capabilities. Some won't have a representation in the dialog.

<Capabilities>
    <Capability Name="internetClient" />
    <uap:Capability Name="documentsLibrary"/>
    <uap:Capability Name="picturesLibrary" />
</Capabilities>

Edit additionally, you can also access some restricted folders by using the rescap (restricted capability) namespace to add <rescap:Capability Name="broadFileSystemAccess" /> to gain access to any file/folder the user has access to.