Tile is essentially non-porous. Grout is the opposite — it's a cement product full of tiny voids, and it absorbs whatever liquid sits on it. That single difference explains almost everything about why tile floors go dingy.
When you mop, the water you're pushing around gets dirtier with every pass. The tile shrugs it off, but the grout lines drink it in. Over enough years, mopping doesn't just fail to clean the grout, it actively feeds it: each session deposits a little more soil into the lines and the surrounding grout haze, which is why an old floor can be mopped weekly and still look permanently grey.
That soil is bound up in grease and traffic film, and it sits below the surface where a mop head can't physically reach. Scrubbing harder with a household cleaner doesn't get to it either. What does work is an alkaline pre-spray that breaks the grease bond, left long enough to do its job, then mechanical agitation and pressurized rinsing that lifts the released soil out of the grout and vacuums it away instead of pushing it to the next tile.
That's usually the surprise for people: a floor they'd written off as stained comes back several shades lighter, because it was never stained in the first place. It was loaded.
The follow-up question is always whether to seal. A penetrating sealer doesn't change how the grout looks — it occupies the pores so spills sit on top long enough to be wiped up instead of soaking in. It's not permanent and it doesn't make the floor self-cleaning, but on a floor that's just been properly cleaned it buys you a much longer stretch before you're back to square one.
In the meantime, one habit helps more than any product: change your mop water when it looks dirty, and give the floor a plain-water pass afterward. Most people mop with detergent and let it dry in place, which leaves a film that soil sticks to.