Matter in the conventional sense is something that is perceivable that occupies physical space - table, pencil, atoms (you can not see them, but they are matter). Two pieces of matter can not occupy the same space. Travelling matter will stop if it encounters other matter, or bounce off - think of rain and an umbrella
Light is comes to your eyes by waves - electromagnetic waves. Multiple waves can occupy the same space, can pass through each other. This is contradictory to the above description of matter
Now...why light is matter!
Back in the day there were explanations of light as a wave - electromagnetic waves and experiments to show it as a wave - e.g. Young's double slit experiment
But around 1900 a combination of thoughts from Planck and Einstein showed that the photoelectric effect and blackbody radiation were due to light in fact being a particle - called a photon
A photon is a small particle of light BUT it has no mass. Einstein showed that energy and mass are linked with the equation E = mc^2, a revelation that suggested that photons that have no mass could be changed into matter and vice versa
Photons (energy) can change to mass as in the reactions going on in the sun. And turning mass into energy is easy...fire
Scientists now realise that light is a wave AND matter and both definitions are not completely correct
Now...why matter is a wave!
Electrons have been shown to produce interference patterns just like waves in the right conditions
This is called wave-particle duality of matter and is central to the idea of quantum mechanics