To put the new tube in the tire, begin by wedging the tube in between the tire and the rim. Make sure that you start with pushing the nozzle through the hole in the rim as shown. Then, continue to push the rest of the tube through the gap between the tire and the rim until the tube is no longer visible.

If the inner tube has a cap over the nozzle you'll have to remove it before you can pull it out of the rim. Just use your fingers to rotate the cap counterclockwise until it comes off.