How to repair the grout on the floor
Looks like Vinyl Composition Tile (VCT). I doubt if there is grout between the tiles. 

The edges of the tiles are chipping because the subfloor is flexing too much. This may be from the subfloor spanning too far or dryrot has decayed the subfloor. 

You’ll need to replace the tile and check the subfloor and underlayment. Before you remove the tile, you should check to make sure it’s not asbestos or the glue is not asbestos. (Most asbestos tiles are 9” x 9” square, but there are other types too.)