Perhaps the easiest way to imagine the 'fluid mosaic model' is like the surface of a swimming pool.
The surface exists because there are things that are incompatible next to each other. In the pool it's water and air. In a cell it's polar stuff (water and things that dissolve in water) and non-polar stuff (fats, lipids, oils, and the like). For a cell, the only place something non-polar is happy inside that two-layered membrane and shielded from all that nasty water.
Some things aren't just one or another, but have a little bit of both. That is mostly what makes up the membrane - molecules with a polar head and a long non-polar tail. Cells use this to their advantage to make large proteins also with polar and non-polar parts so they too bob in the membrane but can have access to things on one side or another of it (and sometimes on both sides).
Like floats bobbing in a pool, things stuck in a membrane aren't necessarily in one spot of the surface or another. Cells keep things anchored when they need to by attaching them to the cytoskeleton, but more often surface proteins just drift around the membrane as they like. This kind of movement by surface proteins in membranes is what first clued scientists in to the idea that the membrane must be fluid.
Really small stuff can often sneak through the membrane on its own. Ions and big things can't make it across without help. And here is where the surface proteins play a big role: some of them act as gates, and because proteins can be made very specific to what fits them, with gates like this a cell can have a lot of control what stays in and what goes out. It can even spend energy to create conditions that would never normally occur because of diffusion and the like.
That's the key notes as far as I can see. Hope that helps!