Why hair dryer can heat up the air (kinetic and internal energies)?

In thermodynamics. the energy of a body is reflected in its temperature.

When a hair dryer heat up its coils, it transfer energy to air molecule via radiation.

One of the ways , but not the most efficient of transferring energy is by radiation, which follows the black body curve, both for the heating element and the air. It is very efficient for air because, as seen in the image below, air has few energy levels to absorb radiation directly and transfer it to kinetic energy .

It is contact with the heating element which will transfer kinetic energy to to the air molecules directly and thus increasing the air temperature. That is why a fan is needed, to circulate the air around the heating element so the molecules pick up kinetic energy and the air becomes hot.

My understand is that Kinetic Energy of air should be relate to the speed of air Internal Energy of air should be relate to the density of air

You should read up a bit on the link.

Then how air molecule carry HEAT?NOTE: assume zero humidity

It is the kinetic energy transferred directly through induced convection of air around the heating element.

There exist radiant heaters without fans ( not hair driers). These work with the radiation photons hitting various solids and transferring their energy to rotational and vibrational levels of the solids , which define the temperature of the solid. We feel radiant heat because of this mechanism.