This word is said to be a Scots colloquial term, though I’ve not been able to track down a reference. It means a place to sit out in, a summerhouse or gazebo, from sit plus oot (a Scots pronunciation of out) plus the noun ending –erie of French origin that’s familiar from words like menagerie and rotisserie.
English newspaper readers suddenly started to see this word during the past summer because it was applied to an art exhibition in the historic landscaped gardens of Belsay House in Northumberland, near Newcastle upon Tyne. A dozen designers and architects were each given a budget and invited to interpret the idea of a sitooterie as a meditation on the perception of landscape. This resulted in intriguing structures, some practical, some more like follies. Whether the exhibition (which has now closed, by the way) will do anything to make sitooterie a permanent part of the English — as opposed to the Scots — tongue, is open to debate.
Several Scottish subscribers have remarked that the word used to have a rather different meaning — a secluded corner where you could take your partner during a dance. It would seem that the word has either shifted sense, or the exhibition organisers have extended its meaning.