What happens with transistor outside max temperature range?

As you may know, semiconductor devices are fabricated doping a very pure silicon (or other, less common, semiconductor materials) substrate using various kinds of ions. Doping different zones of the semiconductor with different types and concentrations of dopants produces the different kinds of semiconductor devices you are accustomed to (diodes, BJTs, FETs) and also (on integrated circuits) resistors and capacitors.

The doping ions give the semiconductor crystal its properties, but they are somewhat intruders in the regular intrinsic semiconductor lattice, since every thermodynamic system at a temperature above 0K tends, if left evolving, to a state of uniform concentration of chemical species. In other words, the ions tend to move away from their position in order to make their concentration in the crystal uniform. This phenomenon is called diffusion and it is contrasted by the forces of the chemical bonds that keep together the crystal.

Note that the bigger the amount of ionic diffusion, the more different regions of the chip lose their "identity ", i.e. their characteristics as electronic devices.

This effect is accelerated by high temperature because the thermal agitation tend to disrupt chemical bonds: ions with higher thermal energy diffuse more easily.

This phenomenon is always present, even at room temperature, but it's usually negligible. Nevertheless, ionic migration is not a linear effect, but an exponential one: so it increases dramatically with temperature. The max temperature listed by manufacturer is a threshold under which the manufacturer can guarantee that the device won't be damaged during the expected life of the part. Over that temperature, all bets are off and ionic migration and other temperature-related effects can actually damage the device in a relatively short time, i.e. the part could have its prospective operating life shortened.

Of course, if the max temp is 175°C and you run the part at 180°C it won't fail at once usually, but it will slowly degrade its performance. The higher the overtemperature, the quicker the degradation.

There are also other effects, though. At high temperatures the tiny wires connecting the chip to the package terminals (bond wires) could get damage from thermal stresses: the materials that make up the component have different thermal expansion coefficients, hence if the bond wire expands less than the surrounding material it may get damaged by excessive mechanical tension, for example. This same mechanism can damage the part at low temperatures (at -60°C you may even have cracks in the package, if you are unlucky enough).