Well, that's exactly the novelty for EM fields. Physicists back in the 1900 believed that light propagated simlarly to a sound wave. They assumed that the space should be filled by an "aether", a fluid that is responsible for transmitting EM waves as air transmits sound waves. This turned out to be wrong.
EM waves travel in the vacuum, at the speed of light, that is the same in every reference frame. They do not need a carrier, they do not need to be represented by dust (or ether) particles smashing each other. Michelson and Morley proved that with their famous experiment, in which they ultimately disproved that the Universe is filled by this stationary aether, that transmits light waves. Basically, they sent out rays of light in different directions of the same length and compared the beams, to see if they presented a phase difference. As the Earth should be moving in this aether, it should experience something like an "aether wind", something akin when you move underwater. In this case, different directions should not be equivalent, and they expected a phase difference of which they knew the expected value.
It turned out that there is no such phase difference. This led physicists to ultimately discard the idea of the aether, and the next step was relativity.
Light propagation in matter can be intuitively associated to sound wave propagation. Basically, the EM field scatters on the charges and dipoles that are present in the medium, and as a result it moves slower. But this is a particular example: light can also propagate in vacuum, and sound waves cannot.
To answer your last question: there is no such "restoring force". An EM wave has definite energy, and while propagating in vacuum this energy is conserved. The field will not go from non zero to zero. Sound waves are "restored" by friction, that dissipates the energy of the moving particles into heat. This does not apply to EM waves: the form of the wave could change with time, but the total energy will always be constant.