If you look at books on (not just Abelian) abstract harmonic analysis, such as Hewitt-Ross, they tell that Tannaka-Krein duality was originally a non-Abelian version for compact topological groups of Pontryagin duality for locally compact Abelian topological groups. The Pontryagin dual is really a group (it consists of all continuous multiplicative characters with pointwise multiplication), and allows one, for instance, to define a notion of Fourier transform on such groups. More generally, the whole topic of duality for locally compact topological groups is a blend of algebra and analysis, just as Schwartz's theory of distributions. Since Tannaka-Krein duality was first formulated in this way, this explains the (harmonic) analyst's interest on the topic.

It was noticed later that the same framework for Lie groups, formulated in the language of Hopf algebras (I believe this was done for the first time in Hochschild's book "The Structure of Lie Groups". Hewitt-Ross's uses the older terminology "Krein algebras"), could be extended to algebraic groups, so the topic also fits naturally within algebraic geometry. Moreover, since group duality essentially tells us that we can recover the group from his representation theory (i.e. its "dual"), one may think of moving that framework to the context of G-bundles, or, more generally, gerbes and stacks (and to even higher categorical contexts). That's what Deligne, Lurie and other people did, it seems to me.

Just a side remark: independently from Deligne's work, there is also another categorification of Tannaka-Krein duality using C*-algebras (more precisely, tensor C*-categories), concluded more or less at the same time as Deligne (after more than 15 years of hard work) by Doplicher and Roberts, in the context of the algebraic analysis of superselection sectors in quantum field theory. This framework applies to precisely the same context as the original Tannaka-Krein duality, i.e. to compact topological groups.