How can wind feel colder than normal air if it has more energy?

Your science teacher was wrong. Wind feels colder to your skin on a hot day because it evaporates your sweat faster, carrying away more heat from your skin than can conduction to still air at the same temperature.

Kinetic energy is equal to 1/2 * (mass) * (velocity^2), so the assertion that kinetic energy is not connected to velocity is also false.

Furthermore, the molecules in air are vibrating around and bouncing off one another far faster (~750 miles an hour) than the movement of the wind. So the additional kinetic energy they have because the wind carrying them is going 20 miles an hour isn't important in this context.

While I am not sure of what "kinetic energy separate from velocity" means$^*$ (I will wait for clarification), there seems to be some confusion as to why wind makes one feel colder, even though the air is moving faster relative to your skin than still air would be.

First, if we are stationary in a location with no wind and air that is at a temperature that is lower than our body temperature, then we essentially will reach a steady-state situation where the air right next to our body is at the same temperature as our body, and the air will become colder as we look farther from our body. Thus, we will essentially develop a "protective layer" of air that insulates us from the colder air around us.$^{**}$

However, if we start moving, or if the wind starts blowing, then this warm protective layer of air will become thinner/non-existent, and the air closer to your body becomes colder, hence you feel colder as well.

It sounds like your science teacher just doesn't know what is happening in this situation, and so they are just saying things that might sound correct until you appear to be satisfied with the explanation so they don't have to be on the spot anymore.

Aside: Something to keep in mind is that humans are horrible thermometers. The classic example showing this is how conductors like metals feel in comparison to insulators like fabric. If we have a conductor and an insulator both at the same temperature, say lower than body temperature, we will feel the conductor to be colder than the fabric. This is not because they are at different temperatures, but it is because the conductor absorbs energy at a faster rate, and so we feel it as "colder".

$^*$ Kinetic energy is defined as $\frac12mv^2$, so I cannot see how kinetic energy can be separate from velocity.

$^{**}$ It is not a perfect insulation, of course, but it is better than nothing.

These other answers are correct, but there is another way to think of it I want to share.

Blowing wind causes forced convection. What this means is that the energy is carried away by the wind. The faster the wind blows, the faster it is able to carry that energy away from you.

Since the heat loss depends on the temperature difference, the faster the wind is able to move that heat away, the cooler the air around you, and thus the higher the temperature difference between you and the air around you; therefore the rate of heat loss is greater and you feel colder.