It is not necessary to go into detail about the form of the potential or any kind of calculation in order to understand this heuristic line of reasoning. My understanding of the statement is the following:
By conservation of momentum/energy, we know that in order to create a particle-antiparticle pair, its total momentum/energy has to be present in some other form. For example, a photon of sufficient momentum can decay into a quark and an antiquark.
As a consequence, whether such a pair is created or not depends both on the energy present and the mass of the particles created. When one just looks at spontaneous (not explicit) symmetry breaking, and this is what Peskin and Schroeder do in the passage you cited, the quark masses are equal to zero (This is also referred to as the chiral limit of QCD). Therefore little energy is required in order to create such a a quark-antiquark pair. Thus, such processes can be expected to contribute to a nonvanishing quark condensate, given by
already at low energies.