Osmosis is, from a scientific perspective is kind of a complicated read - reverse osmosis is even more complicated. However it is a little easier to give a reference to its effect on water quality. It is used on board the International Space Lab to recycle all the H2O fluids - all. There is no water released from the Lab everything including body fluids are reused (drank).
During the reverse osmosis process water is squeezed threw a membrane (they don't call it a filter). The holes in this membrane are 0.0001 of a micron ( 1 ten thousandths) about 1000 times smaller then a human hair. Any of the contaminants we are concerned with are much larger. Bacteria, fluoride, heavy metal would be like trying to drive a large apartment building threw the Detroit tunnels
A lot of people think that a water processor aka reverse osmosis is a bit of an overkill to have in your home. And for some it would be a valid statement. My dad used to drink water from a garden hose. He would also dip his cup in the hole that he cut for ice fishing and drink that like it just came out of the fridge. I couldn't even talk him into a Britta. However, if you are cognoscente of your health and, or believe in maximizing your children's health then it is a very important part of the puzzle.
It is never hooked up domestically by its self. You always incorporate it into a system - a group of filters. That's what makes a reverse osmosis water processing system so effective. It is a series of filters, membrane - and cartridges each dealing with there own specialty.