For loading the chess pieces into your scene, choos "File -> Append", then choose the file you've downloaded, then choose "Groups". Here you will find the Groups "Chess pieces with rig" and "Studio lighting setup". Simply append the chess pieces group, and if you need the extra lighting setup, choose the studio setup group as well.
To replay the chess tournament game, select the armature, then go to the armature settings in the properties panel and choose one of the stored poses by clicking on the "apply specified Pose Library pose to the rig" Button. Do so while the armature is in Object mode. By default all bones need to be selected in Pose mode. If you changed this, only those figurines will move that are selected in pose mode.