What does my teacher mean when he says that all electromagnetic waves travel at the same speed when travelling through a vacuum? If you may, please answer as simple as possible.
Electromagnetic waves include visible light, radio waves, X-rays, and so on. What distinguishes these different bands of light is their frequency (or wavelength). But what they all have in common is that they travel at the same speed in vacuum.
The reason for qualifying 'in vacuum' is because EM waves of different frequencies often propagate at different speeds through material.
The speed of a wave $c$, its wavelength $\lambda$ and frequency $f$ are all related according to $c=\lambda f$. So if $c$ is the same for all EM waves, then if you (say) double the frequency of a wave, its wavelength will halve.
Assume you are walking down the road. You carry a little stick with you. Just for fun you decide to wiggle the stick rhythmically up and down at the rate of one up/down wiggle per second (you are a bit of an olympic expert at stick wiggling so it's very accurate and reliable). Your stick wiggling is at 1Hz. The speed at which you walk down the road is not related to the rate at which you wiggle the stick. You can walk down the road at any speed you like and still wiggle at 1Hz.
Have you got it now ?
The speed of transmission of an e-m wave is not related to the frequency/wavelength.