Let's start with your example. NaBr is an ionic compound. Sodium carries a positive charge and bromine carries a negative charge. NaBr dissolves in water because water is a polar molecule. You can think of one "side" of the water molecule being positive and one side being negative.
Here's where it gets interesting. When you put the NaBr into the water, the water molecules will begin orienting themselves based on their attraction to the sodium/bromine in the molecule. While one water molecule wouldn't be enough to rip the NaBr apart, when you have all of them together, that total attractive force is enough to break the ionic bond between the sodium and the bromine. This leaves a sodium ion and a bromine ion. Once released, these ions don't react; they just float around.
There is a limit though. Like I said, you need more than the force from one water molecule to break the NaBr apart. You may have noticed that if you've ever dumped table salt in water, after a certain point, no more will dissolve. The measure of how much of a substance will dissolve in water is called the solubility.
For the second part of your question, the biggest reason NaBr won't react with water is because of intermolecular forces. Specifically, there is a force called hydrogen bonding, which occurs when a hydrogen atom bonds to nitrogen, oxygen, or fluorine. Water is hydrogen bound to oxygen.
To analyze this potential reaction, think of a chemical reaction as the process of breaking bonds between some molecules (the reactants) and then forming new bonds (the products). You're essentially rearranging atoms.
Hydrogen bonding is the strongest type of intermolecular force. That means that it takes a lot of energy to break the bonds within a water molecule. H2O wants to stay H2O, and that's why no reaction occurs.