Why using smaller portions are better for extractions
Suppose you have 1 liter of water solution of something you want to extract, 1 liter of solvent, and a partition coefficient of 1 (not very practical, but this is just to make the math simpler). You mix them all together, the compound distributes between water and solvent at 1:1 ratio, then you separate the solvent and end up with 50% of your compound. Can we do better than that?

OK, now let's divide the solvent in two equal portions. The first portion of 500 ml takes $1\over3$ of our compound, then the second portion takes $1\over3$ of the rest, or $2\over9$ of the original amount, thus making the total of ${5\over9}\approx55\%$, which is somewhat better than $50\%$.

Believe it or check it yourself, splitting the solvent into more portions will get us even more of our compound. You may call 5% a marginal gain, but with more favorable partition coefficients, the figures may grow quite a while.

And finally, if we split the original solution as well, and apply the portions in the right order, we may get pretty close to the industrial process.