To claim a route, play a set of cards from your hand that match in color, then add your train cars to the board. Once you claim those spaces no one else can take them. Here are some other details.

The color of the cards you play indicates the color of the routes you can claim on the board. However, those gray squares can be claimed with sets of any color.

Also, the number of cards you play must match the number of spaces in the route you want to claim. In the picture here, the four yellow cards will allow the player to claim one of the two routes between New Orleans and Atlanta. That's because four yellow spaces lie between those cities.

Claiming a route earns a player points immediately. See the chart right on the board? Claiming a route of four spaces is worth 7 points.