How does solvent extraction works?
Solvent extraction relies on the relative solubility of a solute in the 2 immiscible solvents. Given the above system, uncharged, organic species will be more soluble in the organic phase, and ionic substances will remain in the aqueous phase.
Given a couple of extractions with fresh solvent, the aqueous phase will retain little if any organic solute. Such solute has been concentrated (partitioned!) in the organic phase. Typically the organic phase is quite volatile: compare the volatility of diethyl ether versus water. The diethyl ether, being quite volatile, may be removed by suction, and the crude organic solute may thus be concentrated and purified (by recrystallization or distillation).