It's incredibly complicated and every type of tissue has it's own specific mechanisms for forming it's 3D structure but the short answer is that there is an "extracellular matrix" that forms a network of macromolecules using collagen, elastin, fibronectin, and a bunch more signaling and structural proteins that organize cell growth.

The ECM starts developing in the fetus and figuring out how it grows from a few cells into a bunch of differentiated organs, muscles, bone, etc. is a core problem in embryology (biology in general, really). If we knew enough to make proper meat, we'd probably be steps away from a Jurassic park scenario.