As far as the rules go you could just look it up (see other answers/comments). I'm going to assume instead you are asking the philosophical/mechanical reasons for this.
One of the key ways to reduce your opponent's score in Ticket to Ride is to stop them making tickets so that they get negative points for them at the end of the game. On the original US map blocking is a very important strategy. You try and guess what routes your opponent is going for and then block them with your own trains. This works best when they are most of the way to connecting two medium routes and you snipe a 2-or-3 carriage route in the middle (e.g. St. Louis to Nashville). At worst it forces them to re-route and potentially complete less tickets overall. At best it can be a game winning strategy, especially late game if they don't have the trains to re-route around the block.
Obviously a good opponent will do their best to throw you off the scent and block you in retaliation.
The double routes help to alleviate the power of blocking strategies, especially down the middle of the US map where you can potentially scupper multiple long tickets in a few good moves. I also really enjoy the meta aspect that can be introduced by double routes: You need to convince other players to block the player who's route is clearly going that way. The fun that is gained from this depends on who you play with so YMMV.
Europe goes even further in allowing people to circumvent blocking by introducing train stations that allow you to use another player's route to complete your tickets. Which of these types of play you prefer will of course affect which version you will prefer. (I personally think Europe is far superior for new players but I am enjoying the US map from a tactical standpoint).
Probably worth also mentioning that double tracks work differently in some of the expansions as they have different numbers of players. On the 2-3 player Switzerland Map:
In 3 Player games, players can use both tracks of the double-routes. In 2 Player
games, once one of the tracks of a double-route is taken, the other one is no longer available.
And on the India Map:
In 4 Player games only, players can use both tracks of the double-routes. In 2 and 3 Player games, once one of the tracks of a double-route is taken, the other one is no longer