Partly because, among all the multiplicative groups of rings

G_n := (Z/nZ)*,

24 is the largest n for which k | n implies that all elements of G_k are squares.

And the largest n such that x² = 1 in Z_n, for all x coprime to n.

Also because of its appearance in Dedekind eta.

Indeed. There are many other interesting properties, including: The number of permutations of four objects; The number of vertices of the 24-cell; The order of the binary tetrahedral group; The order of the group of rotations of the cube; The order of the group of reflections of the tetrahedron; The length of the binary Golay code; The dimension of the Leech lattice; The only non-trivial integer for which 1² + 2² + 3² + ... + n² is itself a square number; The number of Niemeier lattices including the Leech lattice; The number of Kummer's solutions to the hypergeometric differential equation; The number of objects permuted by the largest Mathieu group; The number of unit Hurwitz integers; A constant in Rademacher's formula for the partition function; The exponent in Ramanujan's tau function; The negative reciprocal of the ground state energy of a vibrating string in one dimension; The number of generations required for all exotic elements to disappear in audioactive decay. A large quantity of these are related, in some way or another, to the Leech lattice itself. Sincerely, Adam P. Goucher