Quite simply, heat lightning is the reflection of lightning occurring beyond the horizon; reflecting off the bottom of clouds on the horizon. The light produced, traveling very fast and being relatively unaffected by the air of the bottom-most layer of the Earth's atmosphere, the Troposphere, is able to rapidly traverse vast distances. The refractive properties of the large water droplets and/or ice crystals in clouds can redirect the light's path, so that some of it returns to the Earth's surface many miles away to be perceived by the people there.
Sound waves, on the other hand, are comparatively very slow. They propagate through the molecules composing the air, each molecule "knocking" the next and therefore the sound waves are slowed or dampened in the process. Sound waves can also be blocked to a certain extent by solid substances. A loud noise beyond the horizon is shielded by the horizon, so that only a small residue of the originating sound is able to propagate to the ground beyond line of sight.
Lightning occurs when the disparity between the negative and positive charges in nearby regions of the atmosphere or ground, either horizontal or vertical, overcomes the resistance of the air between them. Lightning can fork or sheet between these regions as an electrical discharge, a flow of the negatively charged electrons, that attempts to equalize the electrical charges of the regions. Sometimes the neighboring fields of electrical charge can be quite complex and the resulting lightning can form glowing three dimensional shapes, most commonly spherical, and therefore called ball lightning. Lightning can occur within clouds, between clouds, between clouds and nearby pockets of ionically charged air, and between clouds and the earth below.
The path of that electrical discharge reaches temperatures of up to 30,000 degrees Kelvin (50,000 degrees Fahrenheit), explosively heating the air. The heated air expands ultrasonically initially, but the generated wave soon decays to a sound wave heard as a series of thunderclaps and rumbles due to interaction with the intervening air and the funneling and echoing caused by the topography (shape) of the intervening land.
The overall result is that the light generated by a discharge of lightning, if reflected by clouds, can travel further than the sound generated by the super-heated air the lightning cooks. This soundless reflection of distant lightning has come to be called heat lightning, because it is most frequently seen on hot, humid, summer evenings. Such weather tends to produce small, localised thunderstorms, thus providing the circumstances most likely to result in clouds on the horizon and lightning strikes beyond.