Inspect the tiles to look for signs of cracking. If multiple tiles show signs of damage, the backer board is likely to blame. The backer board is the material underneath the tiles, and it needs to be treated with a thinset mortar before the floor tiles are set for best results. Replace the damaged tiles and treat the backs of the new ones with latex-fortified thinset mortar to help prevent cracks in the future.
Cut the grout between any two tiles that are pushing against each other or warping. The problem is likely a lack of expansion joints. The tiles move subtly over time, and pressure builds up between them if there aren't expansion joints installed. Making a cut between the affected tiles with a tile cutter alleviates pressure and may save you the trouble of having to install expansion joints after the fact.
Spray discolored tiles with a half and half solution of warm water and white vinegar in a spray bottle and scrub at them with a microfiber cloth to help remove stains and restore the tile without causing damage to it. Palazzo tiles are likely to be damaged if cleaned with harsh solvents, and vinegar removes grease and dirt without stripping the tile of color. Use the solution on the grout as well, and use a toothbrush or scrub brush to help restore the grout color.
Repair minor tile damage with an epoxy to seal the tile back together. This saves you the cost and hassle of replacing a full tile due to a minor crack or chip in the floor.
Seal the floor with a floor sealer to protect it against scratching and additional damage. Sealers fill in the pores in the tile and prevent moisture absorption. Use a mop to apply the sealer. Start at the far wall and work backward toward the door so you don't have to walk through the wet sealer.