from the images it looks like the seams didn't hold, not the structure of the fabric.
Or...it doesn't happen with the new stuff apparently, but could it be an older poly thread that melted when ironed?
What a dreadful disappointment, I'm so sorry Julia. Before I considered repairing, I wash it again, see if any more seams let go. Then you'd know if it was worth the effort. You could cut it down into a table topper or placemats, so not a total loss of time and materials?