What travertine is
Travertine is a limestone laid down by mineral springs, which is why it has the holes. Those voids are usually filled at the factory with a colour-matched resin or cement, and it is the fill, not the stone, that fails first. Over time the fill drops out, the holes collect dirt, and the surface reads as tired long before the stone itself has worn.
How it is restored
The floor is cleaned to lift what is sitting in the open voids, then the voids are re-filled and left to cure. Only then is the surface honed flat and taken through the grits to the finish you want: most travertine looks best honed or satin rather than mirror-polished, because a high gloss exaggerates the fill against the stone. It is sealed with an impregnating sealer once dry.
Indoors and out
Outdoor travertine, common around Sydney pools, needs the same fill work plus treatment for algae and salt. Pressure washing on its own opens the voids further and takes the fill with it, which is how most outdoor travertine gets worse.