So the rule book says that at 1st level the Eidolon gets 4 additional skills to be class skills. Now how do I place ranks in these skills, or what do I do?
You place ranks in skills like any other character. Choosing the 4 extra class skills simply means that when(if?) you do, they get the extra +3 bonus for being a class skill. The Eidolon progression table shows how many skill points it has at a given level.