If you want to listen to whole tracks in the website pages you must be logged in to the Spotify web application, which you can do at this external link (this link will open a new tab in your browser):
You may have to refresh any browser pages that are already open before you will be able to play the complete tracks.
Alternatively you can click on the Spotify icon in the top right hand corner of any playlist, album, or track to open it up in the Spotify application or web application directly. This will enable you to add a playlist into your own collection, add tracks into your own playlists, adjust the player volume etc..