Grout discoloration is said to be removed up to 85%, 15% of the stain left behind is from installation or just exposure over time.
After the grouting process is repeated several times you may consider the option of color sealing your grout as opposed to the tedious
process of constant cleaning.
Allow an alkaline cleaners to absorb into the floors a while it will brake up the dirt and grease. The recommended time is 20-30 min.
Use a grout brush to scrub your floors and be mindful of bumping into your walls, vacuum or mop and then rinse floors with warm water.
Use cotton clothes to dry and then you are ready to apply sealant.