From Moo's Documentation:

Unlike Mouse this module does not aim at full compatibility with Moose's surface syntax, preferring instead of provide full interoperability via the metaclass inflation capabilities described in "MOO AND MOOSE".

This syntax difference is explained in greater detail later on in the Moo documentation, under the section "INCOMPATIBILITIES WITH MOOSE" That documentation leads you to MooX::Types::MooseLike, which leads to MooX::Types::MooseLike::Base. ...and that provides enough information for you to produce the following code, which does work.

package Maze
{
    use Moo;
    use MooX::Types::MooseLike::Base qw(:all);

    has 'maze_map' => (
      is      => 'rw',
      isa     => ArrayRef[ArrayRef[Str]],
      handles => { my_array => 'elements' } 
    );
    has tile_x_dim => ( is => 'rw' );
    has tile_y_dim => ( is => 'rw' );
}

my @drago = ([1,2,3],[1,3,4]);
my $maze = Maze->new( maze_map => \@drago, tile_x_dim => 28, tile_y_di
+m => 36 );
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I made a few other changes too. You're ending lines 5, 6, and 7 with a comma where it should be a semicolon. Also, with Moo, the type isn't wrapped in quotes, whereas it is with Moose.