However,he missed the Dorset countryside so much that he returned there in 1867 and began writing novels and poetry.
what does "that" mean here ?
While I agree with the different possible meanings of that which people have posted, I think Teia has given the most approriate answer when she says that is used here only as a conjunction between two clauses, and it is a conjunction which is often left out. The original sentence can, and often is, written in this way:-
.....he missed the Dorset countryside so much he returned there in 1867......
English omits that very, very often, unlike some other languages.