You probably just need to give your brick floor a good scrubbing to remove the dirt embedded surface layer. The finish on your brick is almost certainly an acrylic floor finish similar to those used on vinyl composition tile floors. What happens is that dirt will become embedded in the relatively soft floor finish, gradually giving it a "greyish yellow look". Amost certainly the grout lines is just where dirt has accumulated, but it's not really embedded in the finish in the grout lines like it is on the brick.
The way you clean these floors is simply to scrub the floor using a floor machine fitted with a rotary brush. That brush will scrub off the dirt embedded surface layer. After a thorough scrubbing to remove that dirty surface layer of finish, you just mop down a new layer of floor finish to replace the dirt embedded finish you scrubbed off.
This really isn't the kind of work you can do yourself without spending a fair bit of money on equipment.
If you phone any of the places listed under "Janitorial Services" in your yellow pages phone directory, virtually all of them would have both a floor machine and cleaning brushes to fit it.
Take a look at the pictures below:
The top picture shows a Centaur Rabbit III floor machine fitted with a wool bonnet and reservoir tank being used to shampoo a carpet.
The middle picture shows a Rabbit III floor machine fitted with a 40 pound weight and a cleaning brush being used to scrub a terra cotta tile floor pretty hard.
The bottom picture shows a Rabbit III fitted with the optional fluorescent light accessory, a reservoir tank and a cleaning brush being used to clean a ceramic tile floor.
Normally, on smooth flat floors, like vinyl composition tile floors, a large nylon "Scotchbrite" pad is used to scrub the dirty surface layer of finish off the floor, and then a new coating of floor finish is applied.
On ceramic tile floors, they do exactly the same thing, only they use a brush instead of a nylon pad. The bristles of the brush get into the grout lines (or mortar joints on your floor) to clean them out too.
There are different kinds of cleaning scrub brushes. Some brushes will have long thin soft bristles, and are used just to polish floor finish. Brushes meant for cleaning will have thicker, stiffer bristles which will scrub the floor much more agressively. The most agressive brushes will use stiff bristles where abrasives have been added to the liquid nylon plastic before drawing that nylon into the long fiber from which the bristles are made. Thus, brushes made from nylon bristles that have abrasives impregnated into them are very aggressive, and are typically used to strip finish off floors.
After scrubbing the dirty surface layer of finish off the floor, they lift the dirty cleaning solution off the floor using a wet/dry vaccuum cleaner.
Then you spray down clean rinse water and scrub the floor a bit more just to dissolve any residual cleaning solution into the rinse water and then they collect that rinse water off the floor with a wet/dry vaccuum.
As soon as they finish lifting the rinse water off the floor, they can put the first coat of new floor finish down. You can typically only buy floor finish by the gallon, 22 liter "enviropac", 5 gallon pail or 45 gallon drum. In your case, I'd recommend you buy a gallon of floor finish and mop it all on in about 4 or 5 moppings. The thicker the layer of floor finish you have on your floor, the better protected the floor is.
Your best bet would be to hire a janitorial company to scrub the floor clean, and then mop a gallon of floor finish onto the floor to provide a new wear layer to last you another 10 or 15 years. Basically, you use a special floor finish mop to mop the finish on your floor. You store the mop head in a clear plastic bag (with no printing on it) so the mop doesn't dry out between applications of finish.
I, myself, prefer "Carefree" from the S. C. Johnson Wax Company, but CastleGuard by the Buckeye Company or just about any high solids floor finish from any national company will give you a hard protective gloss on your floor. Carefree comes in both high gloss and matte. You might prefer a matte finish over brick.
You can rent a floor machine at Home Depot, but it takes a bit of practice to learn how to use it. Also, I doubt you'd be able to rent a cleaning brush from Home Depot. Mine cost me $140, so it'd be cheaper to hire a janitorial company to clean your floor with their brush than to rent a floor machine and buy a brush to do the work yourself.
Any company you hire to clean your floor will give you advice on where to buy a finish mop (if they don't offer to sell you one themselves, and where to buy a top quality acrylic floor finish (if they don't offer to sell you a gallon of the finish they use).
The janitorial service company will see the sense in them cleaning the floor and you mopping on a new coat of finish yourself. The reason for that is that if you wanted them to apply the finish too, then they'd have to send a guy down to your place to mop on a new coat of finish as each coat dries. It only takes a few minutes to put each coat on, but it'll take several hours for each coat to fully dry. So, including the travel time to and from your house, you'll be paying a lot for someone to do something you can easily do yourself.