It's not so easy to see with normal tyres, ones meant for the average person to get from A to B in a practical way. Tyres made for racing can be much softer, these tyres will not last so long but they grip the road much better. If you're really curious, try watching some Formula One races, they will wear out sets of tyres rather quickly and in a relatively small area, so you really get to see it happening.
Tyres will 'grain', the surface appears different (and it's easy to see because they're slicks, not treaded) if they're over-stressed, and little bits come off much like a pencil eraser or rubber. The bits collect beside the racing line, and are easily visible, they're sometimes referred to as 'marbles', and are collectively called clag. They're a hazard, and will cause the car to lose good contact with the track surface.
Weight is a big factor in F1 racing, the cars must weigh a certain amount, and building a car that is below the weight specifications is cheating. Cars are often weighed after the race, and to make sure that the cars are not underweight the drivers will often intentionally drive off the racing line to pick up clag on their tyres, thus adding a bit of weight after racing and immediately before parking and potentially being weighed. If you look at the tyres after the race, before the award ceremony you'll see obvious lumps of rubber stuck to them.
If the track has not been used lately it will change colour during the race. The racing line will become rubberized and darkened, and grip in those areas is increased during the race weekend. Drivers will (sometimes, occasionally) intentionally lay rubber by their pits and where they start the race from to get that little bit of extra grip when they start off.
Here's a picture of extreme clag, you could watch many races and not see this much of a buildup...