Crystals start growing by a process called nucleation. Nucleation can either start with the molecules themselves, which is called unassisted nucleation, or with the help of some solid matter already in the solution, which is called assisted nucleation.
When the molecules are in the solution, most of the time they only see solvent molecules around them. However, they occasionally see other solute molecules. If the compound is a solid when it is pure, there will be some attractive force between these solute molecules. Most of the time when these solute molecules meet they will stay together for a little while, but then other forces will eventually pull them apart. Sometimes the two molecules will stay together long enough to meet up with a third and then a fourth solute molecule.
Most of the time when there are just a few molecules joined together, they break apart. However, once there is a certain number of solute molecules, which is considered a critical size, where the combined attractive forces between the solute molecules become stronger than the other forces in the solution that tend to disrupt the formation of these aggregates. This when the protocrystal or pre-crystal, becomes a nucleation site. As this protocrystal floats around in solution, it encounters other solute molecules. These solute molecules feel the attractive force of the protocrystal and join in. That's how the crystal begins to grow. It continues growing until eventually, it can no longer remain dissolved in the solution and it falls out of solution. Now other solute molecules begin growing on the surface of the crystal and it keeps on getting bigger until there is an equilibrium reached between the solute molecules in the crystal and those still dissolved in the solvent.
Pretty much the same thing happens as in unassisted nucleation, except that a solid surface, such as a stone or brick, acts as a place for solute molecules to meet. A solute molecule encounters the solid surface, it adsorbs to this surface, and stays on it for a certain time before other randomizing forces of the solution knock it off. Solute molecules will tend to adsorb and aggregate on the surface. This is where the protocrystal forms, and the same process as described above happens.