Concerning the bijection G-Bun <-> G(K) \ G(A) / G(O) - it has rather simple intuitive explanation:

1) G-Bun by definition can be described like this - choose covering U_i and consider $G(U_i) \backslash G(U_i \cap U_j) /G(U_j)$ with condition on triple intersection which we do not care for the moment.

2) The main point is that adelic description is particular case of the one above for specific choice of covering. Indeed, choose the following covering: for each point "x" consider its infinitesimal neighbourhood U_x, so regular functions on it are O_x - power series regular at "x" and hence G(O) can be seen as direct product of G(U_x) for all "x". And there is one more chart - "generic point" - regular function on it is "K" and so G(K) is the set of G-regular functions on generic point. Now the intersection of U_x and U_y is empty so we only care about the intersection between U_x and generic point - hence we see G(A) arising as a product over "x" of (U_x G(U_x intersect generic point).

So we get that G-Bun = G(K) \ G(A) / G(O). The triple intersection is empty - so we do not have any condition.

Now about the level structure. My point of view might not be conventional, but it is close to Serre's "Algebraic groups and class fields".

The point is that we can make sense of this level structures via considering the curves with n-cusp singularities.

Let me do only locally. Consider the curve as Spec of f(x): f'(0)=f''(0)=...f'' ''(0) = 0. (for only one derivative we get cusp curve). We might be interested how to describe the bundles on it. The point is that instead of coset G(K) \ G(A) / G(O) : we will have coset G(K) \ G(A) / G(O_{n,0}) i.e. at point zero regular functions are not all power series but such that f'(0)=f''(0)=...f'' ''(0) = 0 .

Now if change "n" and number of points we will get similar cosets and for bundles on singular curves, which might be considered as "level" structures on the original curve.

PS

Sometime ago we played with such curves and bundles on them: http://arxiv.org/abs/hep-th/0309059

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Concerning the bijection G-Bun <-> G(K) \ G(A) / G(O) - it has rather simple intuitive explanation:

1) G-Bun by definition can be described like this - choose covering U_i and consider $G(U_i) \ G(U_i \cap U_j) /G(U_j)$ with condition on triple intersection which we do not care for the moment.

2) The main point is that adelic description is particular case of the one above for specific choice of covering. Indeed, choose the following covering: for each point "x" consider its infinitesimal neighbourhood U_x, so regular functions on it are O_x - power series regular at "x" and hence G(O) can be seen as direct product of G(U_x) for all "x". And there is one more chart - "generic point" - regular function on it is "K" and so G(K) is the set of G-regular functions on generic point. Now the intersection of U_x and U_y is empty so we only care about the intersection between U_x and generic point - hence we see G(A) arising as a product over "x" of (U_x intersect generic point).

So we get that G-Bun = G(K) \ G(A) / G(O). The triple intersection is empty - so we do not have any condition.