Both processes rely on change of solubility of compound in different conditions.
In case of re-crystallization, you first dissolve a compound with impurities in a hot solvent, then cool the solution gradually. Solubility of most solids increases with increasing temperature, so at some point of cooling, your solution will become saturated for a target compound, but still unsaturated for impurities. If you cool it a bit more, target compound will precipitate, while impurities remain in solution.
In case of anti-solvent use, you rely on a fact that compounds have different solubility in different solvent. For example, you have a mixture of two compounds (A and B), that have good solubility in hexane, but one of them (B) also has good solubility in isopropanol, while other (A) does not. You dissolve them in hexane first, then start adding isopropanol. Compound B has good solubility in both solvents, so it will stay in solution at any ratio of these solvent, while solublity of A will decrease with increasing isopropanol content. Eventually, compound A will precipitate.
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