I think many of the other answers boil down to the same underlying idea: Sections of vector bundles are "generalized functions" or "twisted functions" on your manifold/variety/whatever.

For example, Charles mentions subvarieties, which are roughly "zero loci of functions". However, there are no non-constant holomorphic global functions on, say, a projective variety. So how can we talk about subvarieties of a projective variety? Well, we do have non-constant holomorphic functions locally, so we can still define subvarieties locally as being zero loci of functions. But the functions $f_i$ which define a subvariety on one open set $U$ and the functions $g_i$ which define a subvariety on another open set $V$ won't necessarily agree on $U \cap V$. We need some kind of "twist" to make the $f_i$'s and the $g_i$'s match up on $U \cap V$. Upon doing so, the global object that we obtain is not a global function (because, again, there are no non-constant global functions) but a "twisted" global function, in other words a section of a vector bundle whose transition functions are described by these "twists".

Similarly, sections of vector bundles and line bundles are a nice way to talk about functions with poles. Meromorphic functions then become simply sections of a line bundle, which is nice because it allows us to avoid having to talk about $\infty$. This is essentially why line bundles are related to maps to projective space $X \to \mathbb{P}^n$; intuitively, $n+1$ sections of a line bundle over $X$ is the same as $n+1$ meromorphic functions on $X$, which is the same as a map "$X \to (\mathbb{C} \cup \infty)^{n+1}$" which becomes a map "$X \to \mathbb{P}^n$" after we "projectivize".

One way to think of vector bundles and their sections as being invariants of your manifold/variety/whatever is to think of them as describing what kinds of "generalized" or "twisted" functions are possible on your manifold/variety.

The view of sections of vector bundles as being "twisted functions" is also useful for physics, as in e.g. David's answer. For instance, suppose we have a manifold, which we think of as being some space in which particles are moving around. We have local coordinates on the manifold, which are used to describe the position of the particles. Since we are on a manifold, the transitions between the local coordinates are nontrivial. We may also be interested in studying, say, the velocities or momenta (or acceleration, etc.) of the particles moving around in space. On local charts we can describe these momenta easily in terms of the local coordinates, but then for a global description we need transitions between these local descriptions of momenta, just like how we need transitions between the local coordinates in order to describe the manifold globally. The transitions between local descriptions of momenta are not the same as that between the local coordinates (though the former depends on the latter); phrased differently, we obtain a non-trivial (ok, not always non-trivial, but usually non-trivial) vector bundle over our manifold.