Stanley Steamer or another quality steam cleaner will clean that grout. Basic mopping will definitely catch that with most sealers, just keep up on the sealing and the mopping.

It's so odd - I have made this same comment three times in about two weeks.

The Pink Stuff:

Only the paste - not the liquid spray.

Rotary scrub brush:

There are many options - it's nice to buy one that has a bunch of different types of scrub heads/brushes + replacement brushes that you can purchase once the ones that come with the scrubber become worn + IP7 for water resistance.

The two together work really well (plus, they both work on a lot of other things).

Once you have it completely cleaned up, you can add a grout sealant. I don't have a recommendation on that for you - I'm sure that you will receive some suggestions by other Houzz users shortly. If not, go to John Bridge Tile Forum - they have great pros on there who are happy to provide recommendations + advice re: all things related to tile.

Fill a bottle with half bleach and half water and mist it down, then sprinkle some Comet on it. Let it dwell for several hours and the floor will look like new.