An invitation letter is a document by you stating that this family is staying / renting your apartment for 3 weeks.
The reason is that Russian nationals, and other Eastern European nationals that are not part of the EU, or Schengen areas require a visa to the Schengen countries! I mean it is a simple procedure ( aletter from you or the agency that you've used to rent out your apartment), they take it along to the embassy (in this case the Spanish one in Russia), along with this visa, their airtickets or other modes of transportation etc) and apply for the time frame they intend to stay.
the same procedures goes for us who plan to visit Russia etc. I am Norwegian, and I stay at Punta Prima, back home in Norway, but I am often in the Czech Rep and Russia (my girlfriend is Russian) and if visiting Russia, I have to obtain an invitation letter from the hotel I am staying, or an apartment rental, or my girlfreind ofcially inviting me..........................
Hope this could of help!!!