The “Künstlerhaus Gasteiger” in Holzhausen, near Utting. This is not the villa Rilke wanted to rent (that house may have burned down in the 1960s). Image from Wikimedia Commons.
Between 1914 and 1916, Rainer Maria Rilke was involved romantically with the (married) painter Lou Albert-Lasard. In 1915 he found himself somewhat stranded in Munich, waiting to learn whether he would be drafted into the Austrian Army. While there he was consoled by another married Lou, his former lover and life-long good friend Lou Andreas-Salomé (who had a number of other admirers during her lifetime, including Nietzsche and Sigmund Freud). Andreas-Salomé helped him look for a country home to rent on the western shore of the Ammersee. The lake’s western shore and more specifically Holzhausen have long been popular with artists.
Andreas-Salomé: “Yesterday after a bit of an odyssey I rode with Rainer to Holzhausen on the Ammersee so that he could see a small villa owned by Professor Erler. A beautiful, tranquil lakeside park, a charmingly furnished little house seemed to us the clear choice. He’s only uncertain because he would have to commit for the summer months. I think the solitude in nature will do him immense good.”
Image from the Ammersee Kurier
Rilke: “…it is likely that next week I will move into a very small house on the Ammersee, with a housekeeper (whom I am still looking for) and my books. So that I won’t have to speak nor hear and, in a way, be faceless. … the city has become quite unbearable to me”.
“The little house has been taken away from me (when I had just decided this morning), since the Erlers now want to rent out their other, larger villa, not the small one down on the lake!”
“The Erlers”, one assumes, were the brothers Fritz and Erich Erler, both artists, or one of them with spouse. I can’t say which one might have been the professor to whom Andreas-Salomé refers.
Image from Municipality of Utting website
Quotes from “Mich gelüstet’s nach Idylle” by Karen Eva Noetzel (in German).
The Practice Room 
Very interesting. Although I live only a few kilometers from Holzhausen, I never knew that Rilke had planned to rent a house there.