Once you've added all the transitions you want, you have to render the transitions in order for them to work, so go up to the top menu and click on sequence and then hit render all. Sometimes this doesn't work you have to screw around with the playhead and stuff, so just play around with it.

When it is done rendering the red lines will turn purple like the rest of the line.

Once your movie is rendered you can watch it in the preview window with all the transitions in place. An example of a movie with transitions would be..Finalcut Transitions