If we look at word for word translation, Attha means mother in law, Illu means house, and hence, Attha+gaari+intiki=attharintiki.
Gaaru is used to show respect here.
When we add conjunctions to the above words, the word becomes attharintiki.
Dhaari means way. Yedhi means 'where' actually, but with the English usage, it'll be translated as What. [Where's the way = What's the way]
And hence, Dhaaredhi means "what's the way?" [Dhaari+yedhi=Dhaaredhi]
And so, it's What's the way to my Mother-in-law's home.