The airflow ultimately behaves as a function of mach number; airspeed effectively is a function of the amount of airflow at that mach number going over you.
The speed of sound changes with altitude as a function of air-temperature and, with the air going from high to low as you go up, which means the speed of sound goes down (typically under standard conditions from 661.5 to 573.6 knots TAS) which means your indicated airspeed goes down as your true airspeed goes up and with the mach running down you will see your true airspeed for a given mach number get closer together.
First, you'll see the critical mach number appear which is the first point on the wing, the airframe, or both (forgot which) where the local airflow ends up reaching Mach 1. Above this point, the airflow ends up going supersonic, and this area is called supercritical flow. Many jetliners and private-jets see some level of supercritical flow across the top of the wings and, as this supercritical flow decelerates back through the speed of sound, a shockwave forms at the demarcation line between where the airflow transitions from supersonic to subsonic.
Interestingly, since you'll see changes in density where the shockwave is, it can actually distort light which is sometimes visible
Since the formation of the shockwave is formed by the collision of pressure waves (supersonic waves can only go rearward, and subsonic ones can go forward and rearward and at the point where the flow reaches Mach 1, they collide with each other), and this actually drains energy out of the airflow which becomes significant at a value called the drag divergence mach-number. At this point, you'll see an increase in the coefficient of drag (which is the ratio of drag to dynamic pressure) and a reduction in the coefficient of lift (the ratio of lift to dynamic pressure) which starts to exact a penalty on performance (and, while this is sadly not explained as well as it should, the L/D ratio is calculated by simply dividing the CL by CD). The increase in CD is pretty obvious, but the reduction in the CL actually lowers the critical AoA which, by extension also increases the stall speed (The F-86 from what I remember at a weight I don't remember would stall around 114 KCAS at low altitude, but at higher altitude as a function of mach number, it would be around 151 KCAS: This produces a gap of around 37 knots as a function solely of mach-number, and the DC-10 from what I recall would see around 40 knots at low and high mach-number), the effects of turbulence and shockwave position (which progressively moves aft as the mach number increases) can cause aeroelastic effects on the wing and tail (the tail is generally an inverted wing and susceptible to the same problem). This can cause flutter problems to appear which can cause structural problems on it's own and the shockwave's formation also can interfere with the downwash as it travels aft and gets larger (and this greatly contributes to the tails effectiveness) and cause the plane to develop a nose down tendency.
Even if flutter isn't a problem, just the fact that the plane would be in a dive and plunging through the thicker air can be a problem in itself as dynamic forces become considerably higher and with crew trying to frantically recover can make it possible to overstress the wings.
