The concrete area you speak of is probably soil cement. A concrete slab would be prohibitively expensive and they would not have been able to do it in such short time. They may place a thin layer of limestone over the soil cement for final grading. They will then place a plastic drainage layer / padding. It looks like an egg container. It pops together like Lego blocks but its rolled out in sheets.
Once that is done they will roll out the turf and seam it together. It's glued together just like regular home carpet but the tape is wider with a different type of hot melt glue. The backing for the turf has millions of perforations to let the water through. Once the turf is down they get sheep shears and cut out the areas for lines and logos. They glue the lines and logos down with hot melt glue. That makes the lines a little bit higher than the rest of the turf. (It's the same with yard line numbers and hash marks on football fields). That's why you will see players trip on a wide open field.
Once the turf is down they place a layer of sand over the field. They brush it in til about one inch of the blade is showing. The sand is to weigh the turf down so it doesn't blow up in high winds. After the sand they spread rubber pellets over the turf and brush them in until about one quarter of an inch of the grass shows. The rubber is a cushion. The warning track will probably be a little shorter blade with not as much cushion.
If there drainage doesn't work or is inadequate, the rubber pellets will float around and redistribute themselves.
That's the Cliff Notes version.