The reason tires burst at high speeds is somewhat easy to explain.
By the way, why do you Brits call them "Tyres?" They're TIRES!!!!
Okay, past the language barrier
A tire has a flat spot on the bottom, from the weight resting on it. As it rolls (picture this in your mind) the rubber that is flat then moves up and becomes round again. Consequently, the part that was round is now touching the ground and is flat. This is the normal flexing of a tire. But, the problem is that this flexing creates heat. The faster the tire is spinning on the ground, the more it is flexing, the more heat it is creating.
Hot air takes up more space than cold air. When the air inside a tire gets too hot, it becomes "bigger" and causes the tire to explode. That is why a tire on your car is guaranteed to a certain speed- mine are guaranteed to 130 miles per hour, without exploding. It works the same for an airplane. Boeing knows that airplane tires on a 747 can go X speed safely, and they blow up at Y speed. So they tell you- don't go faster than 200 so you don't lose a tire.
Now, a side note, plane tires are not filled with air, they are filled with nitrogen, I believe. The gas is less flammable, so in case they do blow out, there won't be a huge ball of fire. The gas also handles heat better.
That is why tires blow up when you go too fast- they get hot!