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Wick's theorem expresses a time-ordered product of field operators as a sum of normal-ordered products with all possible contractions. Each contraction equals a Feynman propagator. This theorem is the bridge between the abstract operator formalism and the practical Feynman diagram rules.
When computing scattering amplitudes in quantum field theory, you encounter time-ordered products of many field operators -- for instance, <0|T{phi(x1) phi(x2) phi(x3) phi(x4)}|0> in a phi4 theory. Evaluating this directly would require commuting operators through each other, tracking the ordering, and handling the combinatorics of which creation operators pair with which annihilation operators. Wick's theorem reduces this to a systematic bookkeeping exercise.
The theorem states that any time-ordered product can be written as a sum over all possible contractions. A contraction of two fields is defined as the difference between the time-ordered and normal-ordered product: phi(x) phi(y) (contracted) = T{phi(x)phi(y)} - :phi(x)phi(y): = D_F(x - y), which is exactly the Feynman propagator. Wick's theorem says: T{phi_1 phi_2 ... phi_n} = :phi_1 phi_2 ... phi_n: + (all terms with one contraction) + (all terms with two contractions) + ... + (all fully contracted terms). Each contraction replaces a pair of fields with the propagator D_F and removes those fields from the normal-ordered product.
The power of the theorem becomes clear when you take vacuum expectation values. Since <0|:anything:|0> = 0, only fully contracted terms survive. For the four-point function <0|T{phi_1 phi_2 phi_3 phi_4}|0> of a free field, only the three complete pairings contribute: D_F(x1-x2)D_F(x3-x4) + D_F(x1-x3)D_F(x2-x4) + D_F(x1-x4)D_F(x2-x3). Each term is a product of two propagators, and each corresponds to a Feynman diagram with two internal lines connecting four points in different patterns.
When interactions are present, the S-matrix expansion generates time-ordered products with additional field operators from the interaction vertices. Wick's theorem applied to these products produces all possible Feynman diagrams at a given order of perturbation theory. The contraction rules translate directly into Feynman rules: each contraction is an internal propagator, the uncontracted fields connect to external states, and each vertex contributes a coupling constant factor. This is how the intuitive picture of particles exchanging virtual quanta is rigorously derived from the quantum field theory formalism.