The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate parabolic bundles (i.e., vector bundles with flags at finitely many points)? Said differently, how do parabolic bundles arise in nature?
[Math] What are parabolic bundles good for
ag.algebraic-geometryat.algebraic-topologycomplex-geometrydg.differential-geometrygn.general-topology
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I would motivate them as follows: if topological spaces were invented to give a general meaning to "continuous function", then uniform spaces were invented to give a general meaning to "uniformly continuous function". It is clear what "uniformly continuous" should mean for metric spaces and topological groups, but how should the general notion be formalized?
Once this is formalized, one can define the notion of Cauchy net in a uniform space (which is something you cannot do for general topological spaces). This leads to the notion of completeness of course (every Cauchy net converges to at least one point), although the theory is much cleaner for complete Hausdorff uniform spaces, where you have convergence to at most one point as well.
To illustrate this: the Cauchy completion of a uniform space $X$ can be defined in the usual way via equivalence classes of Cauchy nets. It is a complete Hausdorff uniform space $\bar{X}$ together with a map $i: X \to \bar{X}$ which satisfies a universal property: given a complete Hausdorff uniform space $Y$ and a uniformly continuous function $f: X \to Y$, there is a unique uniformly continuous map $\bar{f}: \bar{X} \to Y$ such that $\bar{f} \circ i = f$. (If you omit "Hausdorff" or "uniformly", you lose the universal property, which is arguably the point of the completion.)
The nLab has an article on uniform spaces with some material not included in the Wikipedia article.
There is a bicategory of Dixmier-Douady bundles of algebras which is equivalent to the bicategory of bundle gerbes. In particular, sections into these bundles form algebras.
The price you pay is that the bundles are infinite-dimensional; for that reson I am not sure if that picture persists in a setting "with connections".
I do not know a good source for the bicategory of Dixmier-Douady bundles or for the equivalence. Everything depends certainly on the type of morphisms you consider between the bundles; they clearly have to be of some Morita flavor. You may look into Meinrenken's "Twisted K-homology and group-valued moment maps", Section 2.1.1 and 2.1.4. In Section 2.4 Meinrenken indicates indirectly that his bicategory of Dixmier-Douady bundles is equivalent to the one of bundle gerbes, by transfering the notion of a multiplicative bundle gerbe (which depends on the definitions of 1-morphisms and 2-morphisms) into his language.
Side remark: a bundle gerbe is not the direct generalization of transition functions of a bundle. There is one step in between, namely a bundle 0-gerbe: instead of open sets, it allows for a general surjective submersion as the support for its transition functions. If you take bundle 0-gerbes instead of transition functions, the functor you mentioned at the beginning of your question has as canonical inverse functor. See my paper with Thomas Nikolaus "Four equivalent versions of non-abelian gerbes".
Added (after thinking a bit more about the question): If you want to categorify the vector space of sections into a vector bundle, you first have to fix a categorification of a vector space. An algebra is one possible version of a "2-vector space", probably due to Lurie. Another version, due to Kapranov-Voevodsky, is to define a 2-vector space as a module category over the monoidal category of vector spaces (add some adjectives if you like).
Let us define a section of a bundle gerbe $\mathcal{G}$ over $M$ to be a 1-morphism $s: \mathcal{I} \to \mathcal{G}$, where $\mathcal{I}$ is the trivial bundle gerbe. Then, sections form a category, namely the Hom-category $Hom(\mathcal{I},\mathcal{G})$ of the bicategory of bundle gerbes (the one with the "more morphisms" defined in my paper which was mentioned in the question).
The category $Hom(\mathcal{I},\mathcal{G})$ of sections of $\mathcal{G}$ has naturally the structure of a module category over the monoidal category of vector bundles over $M$. Indeed, a vector bundle is the same as a 1-morphism between trivial gerbes, i.e. an object in $Hom(\mathcal{I},\mathcal{I})$. Under this identification, the module structure is given by composition: $$ Hom(\mathcal{I},\mathcal{G}) \times Hom(\mathcal{I},\mathcal{I}) \to Hom(\mathcal{I},\mathcal{G}). $$ The functor which regards a vector space as a trivial vector bundle induces the claimed module structure over vector spaces.
Summarizing, sections of bundle gerbes do not directly form algebras, but they form Kapranov-Voevodsky 2-vector spaces.
Best Answer
Parabolic bundles were introduced in the 70's by Mehta and Seshadri in the set up of a Riemann surface with cusps. They were trying to generalize the Narasimhan-Seshadri correspondence on a compact Riemann surface (between polystable bundles of degree $0$ and unitary representations of the fundamental group). In the non-compact case, they were able to determine the missing piece of data - partial flags and weights at each cusp. They established what is now called the Mehta-Seshadri correspondence. Then they proceeded to study the moduli space.
Mehta, V. B.; Seshadri, C. S. Moduli of vector bundles on curves with parabolic structures. Math. Ann. 248 (1980), no. 3, 205–239.
https://link.springer.com/article/10.1007/BF01420526
Since then, the definition of a parabolic bundle has been clarified (tensor product with the initial definition is not really computable for instance) and generalized. This is a long story starting with C.Simpson, I.Biswas, and many authors. The upshot is that given a scheme $X$, a Cartier divisor $D$, and an integer $r$, there is a one to one tensor (and Fourier-like) equivalence between parabolic vector bundles on $(X,D)$ with weights in $\frac{1}{r}\mathbb Z$ and standard vector bundles on a certain orbifold $\sqrt[r]{D/X}$, the stack of $r$-th roots of $D$ on $X$. So one can turn your question in: why are these orbifolds natural ? They were first introduced by A.Vistoli in relation with Gromov-Witten theory. They also turned out to be related to the section conjecture (rational points of stack of roots are Grothendieck's packets in his anabelian letter to Faltings). So parabolic sheaves - and stack of roots - are ubiquitous. They are also very strongly related to logarithmic geometry.