The 50Ω standard is basically just convention. There are various stories about how 50Ω came to be chosen. The article Anindo linked is good. There is also The History of 50 Ω or There’s Nothing Magic About 50 Ohms. But the long and short of it is that it is a compromise between low attenuation and power handling.
But it became the standard impedance when designing for transmission line applications way back when. When an IC datasheet says you need to design your PCB traces with a controlled impedance, then you're designing to compensate for transmission line effects. If the impedance of the trace is matched to the output impedance of the IC or source, you reduce the possibility of reflections which would lead to standing waves on the trace and cause all sorts of headaches. Since the designers of the IC are designing with transmission line effects in mind, and since 50Ω is commonly used by convention, the 50Ω standard proliferates.
But 50Ω is by no means special. From this paper on controlled solutions by Advanced Layout Solutions:
Within reason, the absolute impedance value chosen is not normally
important, providing it is controlled along the entire length of the
line. Other constraints in a design often dictate the impedance for
us; it may be chosen based on a design specification (e.g. 65 Ohms for
PCI) or chosen to reduce current (a high impedance). It will generally
be between 45 and 80 Ohms due to typical material geometries, and if
the signal changes layer then the trace geometry should be adjusted
as necessary to maintain a consistent Zo.