One way to understand the symbol of a differential operator (or more generally, a pseudodifferential operator) is to see what the operator does to "wave packets" - functions that are strongly localised in both space and frequency.
Suppose, for instance, that one is working in R^n, and one takes a function psi which is localised to a small neighbourhood of a point x0, and whose Fourier transform is localised to a small neighbourhood of xi0/hbar, for some frequency xi0 (or more geometrically, think of (x0,xi0) as an element of the cotangent bundle of R^n). Such functions exist when hbar is small, e.g. psi(x) = eta( (x-x0)/eps ) e^{i xi0 . (x-x0) / hbar} for some smooth cutoff eta and some small eps (but not as small as hbar).
Now apply a differential operator L of degree d to this wave packet. When one does so (using the chain rule and product rule as appropriate), one obtains a bunch of terms with different powers of 1/hbar attached to them, with the top order term being 1/hbar^d times some quantity a(x0,xi0) times the original wave packet. This number a(x0,xi0) is the principal symbol of a at (x0,xi0). (The lower order terms are related to the lower order components of the symbol, but the precise relationship is icky.)
Basically, when viewed in a wave packet basis, (pseudo)differential operators are diagonal to top order. (This is why one has a pseudodifferential calculus.) The diagonal coefficients are essentially the principal symbol of the operator. [While on this topic: Fourier integral operators (FIO) are essentially diagonal matrices times permutation matrices in the wave packet basis, so they have a symbol as well as a relation (the canonical relation of the FIO, which happens to be a Lagrangian submanifold of phase space).]
One can construct wave packets in arbitrary smooth manifolds, basically because they look flat at small scales, and one can define the inner product xi0 . (x-x0) invariantly (up to lower order corrections) in the asymptotic limit when x is close to x0 and (x0,xi0) is in the cotangent bundle. This gives a way to define the principal symbol on manifolds, which of course agrees with the standard definition.