Left and right is how a stereo file is stored. Mid and side is how it behaves. They describe the same sound, and swapping which pair you work in changes what you are able to reach.
The two signals
Take a stereo file. Add the two channels together and you get the mid: everything the left and the right agree on. That is the kick, the bass, the lead vocal, the snare, anything sitting in the centre of the picture.
Now subtract one channel from the other. What survives is the side: only the parts where the two channels differ. Reverb tails, room, wide pads, the outer edges of a doubled guitar. Anything perfectly centred cancels itself out and vanishes, because a centred sound is identical in both channels and identical minus identical is nothing.
The conversion goes both ways and loses nothing. Turn left and right into mid and side, do something, turn them back, and if you changed nothing in between you have exactly the file you started with.
Why the pair matters
Because a single speaker plays the mid and discards the side. That is what summing to mono does. So mid and side is not a stylistic choice of coordinates, it is the pair that predicts what will happen when your mix reaches a phone or a club subwoofer.
Once you are thinking in these terms, a question like "how wide is this mix" turns into a much more useful one: how much of this mix only exists in the side, and can it afford to be lost?
The four things people use it for
EQ
Cutting the low end from the side alone leaves the bass mono and the top wide. It is the most common mid/side move there is, and it fixes the most damaging mono problem on its own.
Compression
Compressing the mid steadies the centre without squashing the room. Compressing the side does the opposite. On a mix bus the two want different settings, and one stereo compressor gives them the same ones.
Width
Turning the side up widens, turning it down narrows. This is what most stereo wideners are doing underneath, and it is why the aggressive ones cost so much in mono.
Moving energy
Content that lives only in the side can be filtered and summed into the mid, so it is still there after a collapse to mono. The width stays, and the weight survives.
Where to start
The first one on that list, if you only ever do one. High-pass the side signal somewhere around 100 to 150 Hz and the low end becomes mono while everything above it stays exactly as wide as it was. It costs nothing in stereo and it removes the single most expensive thing that can go wrong in mono.
MSBrooo works in mid and side, with separate gain, solo and mute on each, so you can hear the two signals on their own before you decide what to do to either of them.