The gauge field is a way to introduce a local phase shift symmetry into a theory which already has a global phase shift symmetry, by a process called the "minimal coupling prescription."
Say you have a Lagrangian for a free electron and a free photon: \mathcal{L} = \bar{\psi}(i\not{\partial} - m)\psi + \frac{1}{4}F_{\mu\nu}F^{\mu\nu} (where F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu). This Lagrangian has two symmetries: the local gauge transformation of the photon field A^\mu(x) \rightarrow A^\mu(x) + \partial^\mu f(x), and a global phase symmetry \psi(x) \rightarrow e^{i\theta}\psi(x).
Now say we would like the phase symmetry to be local instead--that is, we want to turn the constant \theta into a field \theta(x). If you try to make this transformation, you will see that the Lagrangian is not invariant under it--the problem is that the derivative causes you to pick up the extra term -\bar{\psi}(\not{\partial}\theta)\psi.
We can fix this problem by coupling the two fields together: \mathcal{L} = \bar{\psi}(i\not{\partial} - m)\psi + \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + e\bar{\psi}\not{A}\psi. If you substitute the local phase shift into this Lagrangian, you will see that we can "eat" the troublesome extra term by also making a gauge transformation on the photon field: A^\mu(x) \rightarrow A^\mu(x) + \frac{1}{e}\partial^\mu \theta(x). This causes the interaction term to shift by exactly the same amount as the derivative term, canceling it out. Additionally, because the \frac{1}{4}F_{\mu\nu}F^{\mu\nu} term has a gauge symmetry, it remains invariant under this transformation. Therefore, we have now created a theory which is invariant under a local transformation of both fields together.
Now, you can run this argument the other way--say you have a theory with a free electron only, and you want to make its phase symmetry local. You can do so by introducing a new field, and coupling it to the electron field by the above procedure. Thus, you can also view the minimal coupling prescription as a way to create a photon field out of the air, so that you can make an electron with a local phase symmetry.
Why would you want the electron to have a local phase symmetry? The answer is that if you have a local phase symmetry on the electron, then Noether's Theorem says that there is a conserved current associated with the electron field--that is, the electron has a charge. Since this is an observed property of the electron in the real world, we need to construct a Lagrangian which has that symmetry, and this procedure gives us a way to do it.
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