A polynomial splits over a field K if it can be factored into monomials. For instance, x2+3x+2 splits in the rationals, with factors x+1 and x+2.
Let S be a set of polynomials taken from K[x]. The splitting field for S is the smallest extension that splits all the polynomials in S. Call this extension F/K. Note that F is the splitting field for S over any intermediate extension between F and K.
If U is the set of all roots of all polynomials in S, then F = K(U). If F does not exist, because K is not embedded in a larger field, or that field is not a complete splitting field for S, we can extend the field, so that it acts as a splitting field for S. Adjoin the roots of p(x) one by one, until the field splits every polynomial in S. These missing roots can be "created" out of thin air, using the axioms of ZF set theory, but the details are way beyond the scope of this page. If the set S is uncountable, create the algebraic closure, which requires the axiom of choice. This field splits every polynomial in S; in fact it splits every polynomial. Once all the roots have been brought in, extract the subfield that is minimal and still splits S. Voila, you have created a splitting field.
If p(x) has degree n, repeatedly adjoin the roots of p to show the splitting field has dimension at most n!. For example, try to split x3-2 over the rationals. The real root creates an extension of dimension 3, and a subsequent quadratic extension brings in the complex roots, giving a splitting field of dimension 6.