Henry's Law:

At a constant temperature, the amount of a given gas that dissolves in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid.

Sorry, no five year old will understand that but that's the story in a nutshell. What it means is that if you increase the pressure, you increase the amount of gas that a liquid can hold in suspension. Correspondingly, if you decrease the pressure, you reduce the amount of gas that a liquid can hold. You see a real world demonstration of this every time you open a bottle of beer or soda and release the pressure inside: bubbles appear like magic as the gas comes out of solution. Divers refer to this as offgassing and it's a very serious problem. Fizzing bubbles in your tissues are what causes the bends and that can do horrible things to you, up to and including killing you.

The secret to safe offgassing is to let it happen slowly and gently and to never release the pressure too quickly. You see this when you open a soda bottle very slowly and you don't get so many bubbles compared to when you shake the bottle and pop the top quickly. That's the idea behind safety stops and slow ascents - it allows the surplus gas in the diver's tissues to slowly come out of solution gently instead of fizzing in his bloodstream.

The same thing applies to all the gases in your blood but the difference for divers is that their bodies consume oxygen so they don't have to offgas much/any of it, unlike nitrogen which is inert and all has to be offgassed. That's why divers use nitrox - by breathing gas that has less nitrogen and more oxygen in it they reduce the nitrogen level in their blood which is harder to deal with than the oxygen. There are limits to how well this strategy works but up to that point it works very well.

Trimix is used for a different reason. Both oxygen and nitrogen are narcotic under pressure and if a diver goes deep enough they both will interfere with thinking and muscle control. Helium is essentially not narcotic and is used as a third gas by some divers to help solve this problem. By removing some of the nitrogen (and, in many instances, some of the oxygen) from the gas and replacing it with helium, they avoid the narcosis that can make deeper dives very unsafe. Unfortunately, Henry's law applies to helium, too, and divers using trimix must offgas the helium just like the nitrogen. Even worse, helium is a bad gas when it comes to fizzing in your blood (much worse than nitrogen or oxygen) so extreme care must be taken to make sure that ascent rates are carefully managed, including stops to allow plenty of time for the helium to offgas.