Well, I'll only speak for the one European sport I have extensive experience with.

Cricket's expansion is entirely attributable to the spread of the British Empire. In fact, if you look at the countries that have been the historical titans of cricket: England, Australia, South Africa, New Zealand, India, and Pakistan - they were all under British rule at one point or another. The other countries that round out the more important cricket playing countries are Sri Lanka and Bangladesh, both of whom also experienced English rule. In addition, Cricket is also played in Scotland and Ireland, both of whom are internationally well-represented. In fact, Ireland cricket received a huge boost back in (I believe) the '07 World Cup when they upset Pakistan on St. Patrick's Day. There are many other English former colonies/dominions that play cricket on an international level too, minor ones including: Kenya, Zimbabwe, Canada, etc.

One major exception to this rule is Afghanistan, which actually came across cricket as a result of close contact with Pakistan rather than directly from the British.

The emergence of modern, organised sports took place largely in the late nineteenth and early twentieth centuries. The USA had far less cultural influence than Britain in that period. As already mentioned, soccer and cricket were exported through colonial and trade networks. Rugby was spread to a more limited extent by the same routes.

It is unclear why countries tend to adopt only one code (eg baseball or cricket) but once one becomes dominant it is very hard to install another.

Sources :

David G McComb, Sports in World History (London, 2004)

David Goldblatt, The Ball is Round : a global history of football (London, 2007)