Melting and freezing point

Suppose you have a lump of ice, and you want to melt it to water completely, you will heat it.

As the temperature of the ice reaches $0^\circ \mathrm C$, temperature of the ice will stop rising, and all the heat will be used to convert the ice to water. While this is happening the ice and water will simultaneously exist in equilibrium(because all the ice cannot convert to water instantaneously). After a certain amount of heat is added, all the ice is converted to water.

Notice that this water is at $0^\circ \mathrm C$, because the temperature didn't rise during phase change. The same thing will happen in reverse when you cool down water.

The first process I described is melting, the same process in reverse is freezing. And as temperature doesn't change during melting and freezing, they both happen at the same temperature!

Dumb answer: Because they are the same thing, viewed from different sides. In one direction, it's melting. In the other direction, it's freezing.

It's the same temperature because it's the only temperature at which the liquid phase and the solid phase may co-exist – which is a symmetric description of the temperature.

When we add heat to this mixture of "ice" and liquid, it will keep the temperature at the same point but the percentage of "ice" will be decreasing, and only when all the "ice" is gone, the temperature will stop rising. In the same way, if we remove heat from the mixture, the temperature will be constant for a while as more liquid turns into "ice", and only when the whole body is frozen, the temperature starts to drop.