One intuitive explanation (in the case $R$ regular that Emerton discusses) is that on a derived level a module knows all about the formal neighborhood of its support, and the higher ext's (up to the codimension) account for the missing directions.
Let me assume for simplicity that $M$ is the structure sheaf of a subscheme. Then the functor $Ext(M,-)$ is derived restriction (more precisely restriction-with-supports) to the support of $M$. This functor sets up a derived equivalence (by the derived form of the Barr-Beck theorem, or a fancy form of Koszul duality) between the completion of the regular ring $R$ along the support and a derived refinement of $M$. The idea is that transversally to the support we are using Koszul duality to identify modules over a symmetric algebra and an exterior algebra (ie if the support of $M$ is a smooth variety of codim k, we enhance the structure sheaf of the support by an exterior algebra on a k-dim vector space).
(edit:) More generally, for any $M$ coherent or a perfect complex, ($R$ still regular) $Ext(M,-)$ preserves all limits and colimits on the derived level (in the dg derived catetgory of $R$), and sets up an equivalence between $Ext(M,M)$-modules and whichever $R$-modules are not annihilated by the functor $Ext(M,-)$. For $M$ a structure sheaf this means the formal completion along the support, shouldn't be hard to describe in general. I think it should be easy to deduce from this that we have to have Exts up to the codimension, since we have to account for all the homological complexity of an "open nbhd" of the support using a sheaf of dg algebras supported on the subscheme.