The "X" strap has a small clasp in the centre. Unclip that and spread the two halves open. Place folded sweaters, dress shirts, skirts, and other folded outerwear in the lid compartment. Close the X and zip the cover closed to hold everything in place.
Here's how to prepare a suit & shirt
Edit: I use the supplied flat hanger that comes with the case. Place the hanger inside the suit coat as you would any hanger. Close the suit coat overlapping the sides more than you normally would to button it until the body of the suit coat fits inside the lid. Push the shoulders toward the middle and lay the sleeves on top of the suit coat so it will fit. At this point, the suit is narrow enough to fit. Now close the X and clasp it over the front of the coat. Fold the length of the suit coat in half and zip the cover closed to hold everything in place.
Now, when you close the cover, the apparel in the cover is held in place by the X strap. More can be placed flat on the folded clothing and held there by the zippered cover.
Without the strap and/or the zipped cover, the things in the lid would slide down toward the hinge as you closed the lid unless you had a couple of more hands to hold things in place.
You can use some imagination and find other things they are good for, too.