It's because we're a federation of states whereas you probably live in a country that is some form of a unitary state. Even if you live in one of the few European countries that is also a federation, yours probably doesn't lean as heavily towards being a confederation as ours does. This basically means that our central government and its constituent state governments are considered "mutually sovereign." In some cases the central government has the authority to control how something is handled, while in other cases it is the state government which has the authority.

One of the things which our states have authority over is how elections are conducted within their borders, even the elections for federal offices (within the boundaries of a few federal standards). Unfortunately, this often results in the laws for these being carefully crafted in order to yield a partisan advantage in elections. For example, Republicans are now using voter ID laws in states that they control to obstruct Democrats from voting by passing very stringent standards that mostly target demographics which tend to vote Democrat and also making the acquisition of identification in Democratic areas very difficult.