Why do aircraft need Mach number?
Mind you, not all aircraft need one, especially slower piston ones.

As you get closer to higher speeds, where compressiblity becomes significant, and you are approaching the speed of sound, the air starts to behave differently aerodynamicaly. 

Also: remember how the air is a little bit accelerated over the wing because of the shape of the wing? Well, if you fly close to mach 1 and accelerate the air even further, a little bit of that air is now moving supersonically over the wing. And you now have a shockwave on your wing.

The plane might start pitching down (wing momentum changes), wing stalls (yeap, from too much speed), aerodynamic center moves etc, flight controls might stop working. Google ‘Mcrit’ and ‘mach tuck’ for more, but there are a lot of bad things generally happening in that region.

What it all boils down to is that the wing now has a built-in limitation by the engineers:do not fly faster than (say) mach 0.9 or you will be in trouble.

Now that we have a speed limit that depends on the speed of sound, we really need an instrument to show the pilots when that limit is approaching.