(Edited to correct mistakes signaled in comments below).
I don't know much about the first steps on the theory, Krein and Tannaka. I can just say their work answers a question that seems very natural now, and that I think was natural even then. Since the beginning of the 20th century, representations of groups had been studied, used in many part of mathematics (from Number Theory, think of Artin's L-function to mathematical physics) and more and more emphasized as an invaluable tool to study the group themselves. It was therefore natural to see if a group (compact say) was determined by its representations.
But then, I want to insist on the fundamental role played by Grothendieck in the development of the theory. This role comes in two steps. First Grothendieck developed a pretty complete end extremely elegant theory for a different but analog problem: the problem of determining a group (profinite sat) by its category of sets on which it operates continuously. It is what is called "Grothendieck Galois Theory", for Grothendieck did that in the intention of reformulating and generalizing Galois theory, in a way that would contain his theory of the etale fundamental groups of schemes. What Grothendieck did, roughly, was to define an abstract notion of Galois Category. Those Category admit special functors to the category of Finite Sets called Fibre Functors. Grothendieck proves that those functors are all equivalent and that a Galois category is equivalent to the category of finite set with G-action, where G is the group of automorphism of a fibre functor. He then goes on in establishing an equivalence of categories between profinite groups and Galois categories, with a dictionary translating the most important properties of objects and morphisms on each side. This was done in about 1960, and you can still read it in the remarkable original reference, SGA I.
Already at this time, according to his memoir Recoltes et Semailles, Grothendieck was aware of Krein and Tannaka's work, and interested in the common generalization of it and his own to what would become Tannakian category, that is the study of categories that "look like" categories of representations over a field $E$ of a group (e.g. a pro-reductive group over a field $k$). As he had many other things on his plate, he didn't work on it immediately, but after a little while gave it to do to a student of him, Saavedra. As Grothendieck was aware, the theory is much more difficult than the theory of Galois categories, roughly speaking because of the possibility of the change of base fields and difference between $E$ and $k$ which has no analog in Galois categories. Saavedra seems to have struggled a lot with this material, as would have probably done 99.9% of us. He finally defended in 1972, two years after Gothendieck left IHES, and at a time he was occupied by other, in part non-mathematical subject of interest. Saavedra defined the notion of Tannakian category, but one of his main theorem, the existence of a fibered functor, had, as it was later discovered, a gap.
After that, mathematics continued its development and Tannakian categories began to sprout up like mushrooms (e.g. motives (69, more or less forgotten until the end of the 70's), the dreamt-of category of automorphic representation of Langlands (79), to name two extremely important in number theory). Then Milne and Deligne signaled in 1981 a mistake in Saavedra's thesis, and later with serious efforts, Deligne corrected it.
Modern theory have added many layers of abstraction on that.