Suppose I have an IGBT transistor rated for 60A, that has junction temperature range of -55 to 175C. What happens if I do not exceed max current, but I do exceed max temperature? The characteristics in the datasheets don't seem to have any changes around max temperature limit, so what is the problem? I would like to see some basic physics behind it in the explanation.
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).
Temperature limits in common grade electronics are mostly defined by packaging. Silicon itself has a band gap large enough to operate at temperatures up to +300°C. However, on-chip metallization, wire bonds and case plastic are not designed to withstand high temperatures. They will either deteriorate quickly or crack because of unmatched thermal expansion coefficients.
Another point to consider is the thermal runaway (or, more specifically, secondary breakdown in case of BJTs). Common grade transistors are not designed to prevent it, and can be damaged at high temperature even if the current remains within the spec. Indeed, devices using such transistors are in turn designed without any thermal runaway protection as well, so the current usually does exceed the maximum rating, resulting in magic smoke.
Needless to say, you shouldn't expect the BJT (or any device for that matter) to respect its spec if you run it above the maximum temperature. Its characteristics won't change abruptly when you cross the 175°C boundary, but they will deviate further as temperature increases.
The most dominant effect limiting the temperature range for the use of a transistor is intrinsic conductance.
An undoped semiconductor has some electron-hole pairs. The number of free carriers depends on the temperature and can be calculated with the fermi distribution. Following this an undoped silicium will be a fairly good conductor at 300°C because there are enough free carriers to form electrical currents.
This intrinsic conductance is also present in doped semiconductors, which is understandable considering the few atoms (0,1 - 100 ppm) of dopand between the vast spaces of undoped crystal.
While a transistor's functioning relies on the total absence of carriers in certain areas (depletion zones in pn region) it comes quite clear that this feature isn't working anymore if intrinsic conductance takes place. So at 300°C a Si-BJT or -MOSFET is totally disfunctional.
Intrinsic conductance is a feature present at all temperatures below 0K, however at room temperature the effects of dopand prevail. With rising temperatures the intrinsic conductance grows prominent over the desired function of the transistor until it is rendered unusable. One can anticipate this by looking at the curves in the datasheet depicting the temperature dependend parameters.
For most Si based semiconductors operation above 200°C will lead to high leakage currents which is undesired in most circuit designs.
As noted by @LorenzoDonati degradation of the chip is also an effect to consider. If the whole chip maintains a temperature of 200°C it was not such of a problem. But due to intrinsic conductance in the semiconductor so called hot spots tend to form. These are somehow runaway regions heating up faster than the rest. This leads to local overtemperature with accelerated diffusion processes and eventual destruction of the transistor.
The lower temperature boundaries are due to the different expansion coefficients of chip, leadframe and housing which may crack the device when cooling it too far.