First let's do a scalar field. A classical real scalar field assigns a real number to each point in spacetime. All observers agree on the value of that number for each point. If Alice uses coordinates x, she can write down a function [itex]\varphi(x)[/itex] that gives the number assigned by the field to the point she labels with coordinates x. Bob uses different coordinates x', related to Alice's by a Lorentz transformation, x'=ax. Bob also uses a different function, [itex]\varphi'[/itex], of his coordinates. However, since Bob and Alice agree on the value assigned by the field to any particular point, the numerical values of Bob's function of Bob's coordinates must agree with the numerical values of Alice's function of Alice's coordinates; that is, we must have [itex]\varphi'(x')=\varphi(x)[/itex].
For fields in other representations of the Lorentz group, such as a Dirac field, this gets generalized to [itex]\psi'(x')=S(a)\psi(x)[/itex], where S(a) is a matrix that acts on the index carried by the field.