As an European, I find it very strange for a modern democracy to allow someone to obstruct the functioning of a democratically-elected political organisation. So, is there a historical (or other) reason why this is allowed in the US and not in most other countries?
The Senate must close the debate in order to proceed. In the original rules, this required a motion called the previous question. In 1805, Aaron Burr argued that this was unnecessary and the rules were changed such that it required unanimous consent that debate was over and they were ready to vote. Until 1837, they never denied consent.
From 1837 to 1917, one or more Senators could block cloture of debate by withholding consent. This power was used rarely but enough that the rules were changed in 1917 to allow the Senate to vote to close the debate. After that, it required a supermajority to close debate. This was primarily for historical reasons. They were accustomed to being able to block debate this way and so left it.
Traditionally, the Senate has been a more collegiate body. Its members were more moderate and less partisan. As a result, this system usually worked. Over time it has been used more and more frequently, the size of the supermajority has been reduced, and the times when it can't be used have been expanded (budget reconciliation, nominations other than to the Supreme Court). In these increasingly partisan and ideological times, it seems less and less like another reason for moderate consensus and more and more like an obstacle.
To summarize, the filibuster started as an accident of the Senate rules. It grew into a way for minorities to block contentious legislation. When it really gets in the way, the majority tends to reduce its power. And that's where we are now: a supermajority of sixty votes is required for legislation and important nominations.
In the United States Senate, most filibusters fall into two categories:
A large minority objects to a nomination or bill. (Usually, the nomination or bill has regional or national importance. For example, a Supreme Court nomination.) This kind of filibuster effectively causes the Senate to require 60 votes (instead of 51) to pass many kinds of laws, or approve many nominations.
A single senator objects to a nomination or bill. (Usually, the nomination disproportionately affects the Senator's state, and/or the Administration has not cooperated with the senator on a request the Senator made.) While this kind of filibuster can also be overridden by 60 votes, usually it is not. In practice, if a state has a senator of the president's party, that senator has veto power over federal judges (or other appointed officials) that will decide issues in the senator's state. If neither senator from a state is from the president's party, then either of them might object to appointments that disproportionately affect their state.
The latter effect is an "emergent property" of the Constitution, but I do not think that the Founding Fathers would have objected to it. When there is doubt about which judges would be best for different parts of the country, the tendency will be to choose judges whose opinions tend to match the region they are in. If different regions' appeals courts reach different conclusions about a legal issue, the Supreme Court can set a precedent to standardize how that legal issue is handled.
On a related note, students at the military academies are nominated by individual senators and congressmen.