The radiation you refer to is carried by particles that have "zero rest mass", which means they would not exist in any reference frame in which they were not moving. Since existence is not allowed to depend on reference frame, that means there is no reference frame in which those particles are not moving. That also means they must move faster than any allowable reference frame, i.e., there is a "speed limit" for observers. Other arguments indicate that all particles with zero rest mass must move at this same speed limit. Now, that speed could have been infinite, but that's a very strange universe where things can move infinitely fast! Ours doesn't have that property, so the speed of light is finite. Why that speed happens to be 300,000 km/s, no one has any idea.