Why is solvent extraction important?
You have performed your reaction, and have worked it up in water; your drug, your compound, is in water (it might be present as an oil or as a dispersion).
So how do you purify your drug? You could remove the water. However, water has a high boiling point, and it is notoriously difficult to remove by distillation.
An alternative means, would be to wash the water solution with a water immiscible organic solvent. Chloroform, methylene chloride, and diethyl ether are typically used as such solvents. So you give your water solution a good wash (or several washes) in a separating funnel. Each time you perform an exraction, the organic extract is reserved, and finally, you have a volume of organic solvent, with whatever it is you are trying to isolate dissolved in this volume.
Now the organic extract (so-called) still has to be dried, and still we have to remove the solvent from whatever it is you are trying to isolate. How does this help you? Well, the organic solvents are quite volatile, and may be stripped off on a rotary evaporator. You are left with the crude compound, ready for distillation or recrystallization.
Another scenario, you are in the perfume caper, and wish to extract the scent in roses to add to your fragrance. You grow a field of roses, harvest your blossoms, and extract the blossoms with a suitable organic solvent. Now you have an organic solution of rose blossom essence. Of course, the solvent may be removed, and the essence used.