Maverick asked me the other day what connection could possibly be between the theory of direct reference and existence. Well, there is certainly a connection between direct reference and the verb 'exists'. If the direct reference theory is correct, then this verb cannot take a singular term as a subject. So we can say 'An American philosopher exists', meaning that 'philosopher' is truly predicated of at least one singular term referring to an American, (e.g. 'William Lane Craig'). But we can't say 'William Lane Craig exists', because it is ill-formed. We can predicate 'exists' of general terms only. See the argument I gave here.
The direct reference theory is not to be confused with 'linguistic idealism' – whatever that is. The theory does not deny there is any 'extra linguistic reality'. It simply denies that 'William Lane Craig exists' is meaningful, in the strictest sense of 'meaningful'. If it means anything, the sentence means that the proper name 'William Lane Craig' refers to something. But if it referred to nothing, it would not be a proper name – for the direct reference theory says that whatever counts as a proper name must be meaningful (as opposed to a string of letters or an utterance), and that its meaning is what it refers to. Therefore if the utterance refers to nothing, it means nothing, and so cannot be a proper name. And the fact it refers to something guarantees that it refers to something in 'extra linguistic reality'.
As far 'existence', which is an abstract noun formed from the verb 'exists' – well, 'The existence of William Lane Craig' presumably alludes to the fact that 'William Lane Craig' refers to something. Simple.