An artificial satellite orbiting the earth at an altitude of a few hundred kilometers loses a small amount of energy as heat due to friction of the thin atmosphere at that altitude. Why then does its speed increase progressively as it comes closer and closer to the earth?

As a satellite comes closer to the earth, its potential energy decreases. According to the law of conservation of energy, the total energy of the system must remain constant.
Now,  total energy = KE + PE + heat energy
Since the heat loss is very small, a decrease in potential energy results in a net increase in kinetic energy. Hence the speed of a satellite progressively increases as it comes closer and closer to the earth.