Practitioners of sacred geometry hold that contemplating and constructing these forms attunes the mind to an order underlying appearance, and that building with them does the same for a space. That is a claim about meaning and attention, and stated in that form it is not the sort of claim that admits of testing.
The practical tradition is real and documented. Gothic masons worked from geometric schemes, ad quadratum and ad triangulum constructions, in which a whole elevation is generated from a square or a triangle, and the surviving lodge books show it. Islamic geometric ornament developed a mathematics of periodic and quasi-periodic tiling several centuries before it was described in the West, and the girih patterns of medieval Iran and Anatolia are among the most sophisticated ornamental geometry ever executed. Yantras and mandalas in Indian and Tibetan practice are constructed to precise rules and used as supports for meditation.
In modern esoteric practice the working forms are a smaller and more recent set. The Flower of Life, a hexagonal lattice of overlapping circles, appears as ornament in the ancient world, but its modern name and its elaborate interpretive system come from late twentieth-century esoteric writing, principally that of Drunvalo Melchizedek from the 1990s. The Metatron's Cube figure and the association of the Flower of Life with the Platonic solids belong to the same recent layer. Vesica piscis constructions and the golden spiral are used in design and meditation.
The honest summary is that the geometry is old and shared across cultures; the specific interpretive apparatus now sold under the name sacred geometry is largely a modern synthesis; and the mathematical content is more interesting than most of the claims made on its behalf.