Verified answer

The "bridge" is the part where your strings attach. It's mounted to the surface of the guitar. From your description of looking inside the guitar, I think you're seeing the X-bracing on the underside of the top. Some makers use a cloth tape on the joints and/or ends of the bracing. I imagine it helps make the glue joint stronger, and insures that the braces don't rattle. Anyway, my $2000 Martin guitar has the same thing, so don't judge the quality of your guitar by the tape.

It's perferctly normal. Some guitars have cloth tape running across the grain on the inside of the sides too.

The tape helps to mute the ringing produce by the harmony table just like putting a pillow in a Bass drum to kill the boominess.