Dear Malena!
The "its" refers to the White House... the wives have redecorated the White House's (=its) rooms.
The meaning is, they arrived at the house, and wanted to make sure everyone notices that the Clintons or Kennedys have "taken over" the house. So they decorate the house's rooms. I'm sure they did decorate their private rooms, too. But the important thing here is that they wanted to show the public that they were at home now there and staying for the next 4 or 8 years. So they probably put up some family portraits or art that they own in public halls and stairways, or put a new carpet in the Oval Office :p whatever presidents' wives do if you let them ;)
As I said, "their" is grammatically and also in its meaning a possible word to replace "its" with. But "its" should be used when talking about all of the house's rooms, and not just the wives' private ones.
I hope this answers the question, if not please let me know! :)
Okay, You're absolutely right, tigerwoman