Only three regular polygons tile the plane by themselves: the equilateral triangle, the square and the hexagon. The proof is immediate. Copies meeting at a point must have interior angles summing to 360 degrees, so the interior angle has to divide 360 exactly. The interior angles of regular polygons run 60, 90, 108, 120, 128.57 and upward, and from 120 onward nothing divides 360 again. Only 60, 90 and 120 work.
Among those three the hexagon is the efficient one, and the claim that it is has both a name and a date. The honeycomb conjecture states that a regular hexagonal grid is the way to divide a surface into regions of equal area using the least total perimeter. Varro states something close to it in 36 BC and it is often credited to Pappus in the fourth century AD, which makes it one of the oldest conjectures in mathematics. Laszlo Fejes Toth proved it in 1943 for the restricted case in which every cell is a convex polygon. The general case, allowing cells with curved boundaries of any shape at all, resisted until Thomas Hales proved it in 1999. The key step was showing that the advantage a cell gains by bulging outward never outweighs what its neighbour loses by bulging inward.
Two cautions belong here. First, the theorem concerns dividing a plane, not honeycombs; the three-dimensional question of the most efficient way to close the end of a honeycomb cell is a different problem, and the arrangement bees actually use is very slightly less efficient than the best known. Second, real honeycomb hexagons are partly the result of wax flowing and settling under the bees' body heat rather than being built hexagonal from the outset, and how much of the pattern is behaviour and how much is physics is still investigated.
Kelvin's related question, the best way to partition three-dimensional space into equal-volume cells, remains open. Kelvin's own answer of 1887 stood unbeaten until Weaire and Phelan found a better one in 1994, and nobody has proved theirs optimal either.