Largely because
The borders were designed around ethnic lines (i.e Czech Republic and Slovakia, some countries of the former Yugoslavia) in the first place. By contrast borders constructed in, for example, the Middle-East ignored existing ethnicity boundaries. Iraq is probably the best example of this.
Mass ethnic cleansing at the end of WWII to make ethnicity fit desired borders: millions of Germans were forcibly removed from modern day Czech Republic, Poland and Kaliningrad to make way for settlers of other ethnicity.
Because European states were particularly successful at forging national identities: Italy and France are the two best examples of this. Prior to the unification someone from Sicily or Naples would not have spoken Italian (which was a derivation of the language of the Piedmont region) and would have thought of someone from Milan as foreign as a Frenchmen. Deep divisions between northern and southern Italy still exist today. France was also a mish-mash of ethnicity which greatly differed from each other in 1789. But the French government during the 19th century was able to use public schools and the myths of the revolution to forge a cohesive French national identity.
Those countries are not necessarily as homogeneous as you might think, the UK for instance, still has 4 distinct ethnic groups within it (Irish, Welsh, Scottish, English). Northern Ireland suffered various sectarian and ethnic conflicts all the way up to the 1990s. Belgium is composed of two distinct ethnic groups: the Flemish and the Wallonians which might yet break up the country one day. Spain is still struggling with the ethnic enclaves of Basque and Catalonia, and of course you've probably heard of the issue of ethnic Russian minorities in Ukraine over the last couple of years or so.