Like for sheaf of groups, you can try to describe the patching data required to glue sheaves of groupoids ("stacks"). A grebe is a sheaf of groupoids, locally equivalent to a sheaf of the form $BG$ for a sheaf of groups $G$. However, since $Hom(BG,BH)$ and $Hom(G,H)$ are not the same, gluing these local $BG$-s together require different descent datum than what you need in order to glue the $G$-s thenselves. In some sense, it is "easier" to glue the classifying groupoids. In fact, $Hom(BG,BH)\cong Hom(B,H)//H$, where here the quotient is the orbit groupoid. It has $\pi_0Hom(BG,BH) = Hom(G,H)/H$ and $\pi_1(Hom(BG,BH),f)=Z_H(Im(f))$, the centralizer of the image of $f$. The gluing data for a grebe takes into account both $\pi_0$ and $\pi_1$. When trying to glue the local $BG$-s in pairs, we need to choose a homotopy class of isomorphisms between their restrictions to the intersection. Namely, for $U_1$ and $U_2$ we need a class in $\pi_0(Hom(BG_1,BG_2))$. But now the compatibility of the choice become an extra structure rather than a property, and the collection of choices of compatibilities of this identifications at triple intersections becomes a torsor for $\pi_1(Hom(BG_1,BG_3))$ where $BG_i$ the the local stack at the open set $U_i$ for $i=1,2,3$. Hence, essentially, the $\pi_1$ controls the degrees of freedom in choosing how the gluing data is compatible. Of course, now this data has to strictly satisfy compatibility on quadraple intersections.
From this description of the descent data for a stack, you immediately see that the band is just what happen when we ignore the $\pi_1$-part. Equivalently, it describes how the local $BG$-s glue when considered as objects of the 1-category of groupoids, where we identify naturally isomorphic functors. While the glued object is not very meaningful, it still gives some partial information about the stack itself, more or less like the components of a top. space gives information about the space itself.