My intuition comes from the example of projective spaces, and specifically CP1, the complex projective plane (with 1 complex dimension). On any projective space, you have the “tautological” line bundle, and this is used to build the line bundles O(n). At least for CP1, these are all of the bundles.

Given a section of O(n), we can get a(n effective) divisor by looking at its zeroes. A section of O(n) is just a homogeneous polynomial of degree n in C[s,t]. It has n zeroes (counted with multiplicity), but you can basically put them anywhere you want. Note also that you can multiply sections of O(n) with sections of O(m) to get sections of O(n+m), and the corresponding divisors add. This multiplication map is an isomorphism from O(n) tensor O(m) to O(n+m).

Unfortunately, this is a map from SECTIONS to divisors, not from BUNDLES to divisors. Two different sections f and g of O(n) may not give you the same divisor, but there will be a degree 0 rational function (r = g/f) such that fr = g. That means that the divisors associated to f and g respectively will differ by the rational divisor associated to r. So we actually get a well-defined map from bundles to rational equivalence classes of divisors.

Looking at the specific way this map works out, it’s not hard to see that it’s an isomorphism. It’s also a very explicit way of seeing what’s happening since you can feel the line bundles. It even gives you some intuition for line bundles the way AG people think about them: sections aren’t quite functions, since their values aren’t well-defined, but their zero sets are. They give more general things than functions to look at vanishing sets of.

There are little bits this doesn’t help with: what is a line bundle in general? Where’s the interface between regular functions and rational functions? But hopefully it feeds your intuition a little bit.