A signed distance field answers one question everywhere in space: how far to the nearest surface? Positive outside, negative inside, zero exactly on the skin. A sphere’s SDF is one line, distance to center minus radius, and a library of such formulas exists for boxes, tori, capsules, and stranger things.
The power is in combination. Take the minimum of two SDFs and the shapes union; the maximum intersects them; a soft-minimum melts them together like mercury, the smooth blending that gives SDF work its liquid, organic signature. Repeat space with a modulo and one object becomes an infinite lattice. Twist, bend, and warp with a few operations on the input point. Modeling becomes algebra, and raymarching renders the result directly.
SDFs also solved a flatter problem: text and icons. Valve popularized storing glyphs as distance fields so they scale crisply at any size from tiny textures, and UI rendering uses the trick everywhere. In audio-reactive work the appeal is the same as everywhere else in the shader world: every constant in the distance function is a knob, and music can turn all of them.