1. How does making a different profile allow one to play Skyrim without breaking the game as the save games progress?
This is done by MO's use of a virtual file system (VFS) rather than placing the files directly into the game's Data folder like other mod managers do. The VFS allows one to create several different profiles with different mods installed in each profile without messing with anything in the actual game's folder structure. The VFS is presented to the game at launch and only the "winning" files are presented. Therefore, if you have two mods which both provide file "xyz.dds", only the file with the higher priority (lower on the mod list in the left pane) will be presented to the game at runtime.
2. In other words, can one create different profiles as the game progresses by adding mods and deselecting mods, then selecting and deselecting mods to create different profiles and essentially being able to try out different mod builds and different mods?
Yes, but doing so on the same save will still present issues just like it would with any other mod manager or manual installation. Each MO profile should have it's own, separate save (you can prevent cross contamination of saves by turning on "local savegames" so you will only see the saves for that profile). However, one way around this is to create a "vanilla" save. That vanilla save can be used across all profiles (keep "local savegames" off for this). For example, start a new game with only the mods stalled that would be basic to all your profiles which you would never uninstall (such as SKSE, USLEEP, UHRP, etc). Then play through the Helgen intro and save after the cave exit. This save would be able to be used on all profiles.
3. How do people safely test out all kinds of mods as their game progresses without breaking their game?
I think this is getting to the root of what you're asking.
When testing out a new mod on a play-through, you will create a new save before installing the mod. Then exit the game and install the new mod for testing. After installation, load the game up and play for a while to test the new mod. I wouldn't recommend doing anything that you will regret losing and causing you to play through an area again because if you decide you don't want to use the mod, then you'll lose all that work. If you decide you don't want to keep the mod you're testing, uninstall it and reload the save you created just before installing the new mod. This keeps your saves clean so you can continue playing as normal (it'll be like you never installed the mod).
Keep in mind, this is only necessary for mods which have plugins and scripts. Mods only containing meshes and/or textures can be installed and uninstalled at any time without doing any harm to your saves.