I saw a shooting star as I was standing on the roof of my home. My home is double storey. And the height of that shooting star that I saw was maybe 10 to 13 meters. It was burned down at that height. My question is why haven't the shooting stars burnt completely when they enter the earth's atmosphere where the air is thick? Why sometimes a small piece gets to come very close to the ground and then burns?
What you described is simply impossible. Meteors glow because its surface is being vaporized. When they enter the upper layers of the Earth's atmosphere they are typically travelling several kilometers per second (that's thousands of mph). They are going so fast that the air in front of it doesn't have time to get out of the way of the hypersonic meteor, so the air gets compressed. Compressed air naturally heats up, and because the air can't get out of the way, it stays very hot. This heat will vaporize the leading surface of the meteor (the "front", so to speak), making it glow very brightly. It also leaves a trail of plasma, which is the streak you see behind it.
As the meteor travels lower and lower into the atmosphere, it will slow down. Once it gets to about 15 km (9 miles up), the meteor will have slowed down to below 2 km/s (4000 mph). This speed is too slow to vaporize the surface. Thus, the meteor stops glowing red hot, and is essentially invisible from then on. Eventually it will slow down to about 150 m/s (300 mph). That's only a bit faster than terminal velocity of the average human, and slower than a jet plane. Eventually it will hit the ground at roughly the same as ambient temperature (this vaporization process is very heat efficient).
You claimed that you saw the meteor 10-13 m above your head, but this would not be possible. In order to be glowing, it would need to be travelling at least 2 km/s (4000 mph), which is not possible for a meteor as it would have already slowed down in its flight.
I think you may have been mistaken. Distance is generally hard to measure. In fact, the ancients did not know if the Sun was close by and fairly small, or far away and fairly large. The same went for the Moon. It was only until the Greeks figured out how to measure the distance using some neat trigonometry that we got the Moon's size distance and size. They also got the Sun's size and distance but it was pretty far off, though the way of measuring was correct.