Why is solar energy transferred to earth by radiation

Because between the Earth and Sun there is vacuum

There are three ways in which heat is transmitted:

1.By Conduction, when the transmission is by the direct contact.

2.By Convection, heat transfer in fluids (like water or the air, for example).  

3.By Radiation, by the electromagnetic waves (they can travel through any medium and in vacumm) .

So, the space between the Earth and the Sun is vacuum, this means the energy cannot be transmitted by convection, nor conduction. It must be transmitted by electromagnetic waves that are able to travel with or without a medium, and this is called radiation.

what experimental evidence led scientists to change from the previous model to this one?

a few of the positive particles aimed at a gold foil seemed to bounce back.

the colors of light emitted from heated atoms had very specific energies.

experiments with water vapor showed that elements combine in specific proportions.

cathode rays were bent in the same way whenever a magnet was brought near them.