If you look at books on (non-Abelian) abstract harmonic analysis, such as E. Hewitt and K. A. Ross's *Abstract Harmonic Analysis, Volume II* (Springer-Verlag, 1970), 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 $G$. The Pontryagin dual $\widehat{G}$ is also a (<s>unfortunately, not necessarily</s> locally compact - **edit (June 15th 2022)** as pointed by KConrad in the comments below, see also e.g. Theorem 23.15, pp. 361-362 of E. Hewitt and K. A. Ross, *Abstract Harmonic Analysis, Volume I* (2nd. edition, Springer-Verlag, 1979)) Abelian topological group (it consists of all continuous multiplicative characters with pointwise multiplication, endowed with the compact-open topology), and allows one, for instance, to define a notion of Fourier transform in $G$ using its Haar measure (which is just a multiple of Lebesgue measure in case $G=\mathbb{R}^n$). 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 compact 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 its 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.