I think everyone touched on a piece of it, but nobody got it 100% right. A fine arts academy in any province will add 5/month to your stability investments. In other words, if you allocate 20 ducats/month to stability, with an academy you'll be getting 25 (not counting monarch and religion bonuses, of course). With 2 academies it's 30, etc.
A fine arts academy in your CAPITAL province will add a 12 ducat bonus to your year-end income (the large lump sum of cash you get every january 1st). This sum is cash at the end of the year; it is not anything to be divided between stability or research. A fine arts in any province other than the capital will add 6 ducats every year instead of 12.
The third effect of fine arts academies is to increase tax/production incomes of a province it has been built in. When you click on a province, you get 2 numbers, remember? The right one is the value generated by the province in its center of trade; the left one is the tax/production income of the province, which a manufactory will increase substantially. This is what directly increases your country's monthly income, the stuff you can allocate between cash/stability/research with the sliders.
