When you recrystallise in a lab, you dissolve all the reaction products in hot solvent, then cool the solvent down so the product you want crystallises.
The idea is, the impurities should be soluble at high and low temperatures (hopefully) so they stay in solution as you cool down, but your product should be soluble at high temperature but a lot less soluble at low temperature. So if you have a saturated solution (ie a solution that has the maximum amount of product dissolved in a certain volume of solvent) at high T then when you cool down lots of product crashes out (which just means it precipitates to form a solid).
So the impure product should be dissolved in hot solvent, because thats when it ismost soluble, just using cold solvent would result in a solution that, when cooled with ice, would lead to very little product being precipitated out.
This is why a big difference in solubilities at different T is important, if it dissolved equally well at high T and low T, then once you'd dissolved it, you wouldn't be able to get the solid to precipitate!
As an example (using the number you're given) lets say I dissolved 56.3grams of benzoic acid in 1litre or 100
oC water, that would be saturated, then when i cooled it to room T (25
oC) only 3.4grams would stay in solution, so id have 52.9grams of purified product, that's pretty good yield.
If instead I used a solvent where i could dissolve 56.3 grams per litre at 100
oC, and 50.0 grams per litre at 25
oC, then cooling would only give 6.3 grams of purified product which is rubbish yield!
so the difference in solubility with T is crucial, as is dissolving in hot solvent (the hotter, the more soluble which means the more crashes out when I cool).
The third point, washign with warm solvent, defeats the point of recrystallising because your product is more soluble in warm solvent so you'll just wash away product you worked hard to keep. Instead using ice cold solvent means the the product isn't washed away so much, again giving better yield.
Sorry for the long post, hope it clears things up