My answer will be brief unfortunately. heres how I think about it. Electrons in the band that is near the Fermi energy interact through some attractive force. The ground state of this system is the superconducting state, with all those electrons paired up and you have a superconducting condensate which is that "cooper sea" Now we consider low-energy excitations of this state. Turns out this just come about by breaking one of this cooper pairs (by thermal excitation or otherwise). This costs an energy equal to the superconducting gap. So those quasiparticles, which are just the elementary excitations of the ground state (creation operators acting on the vacuum state), can be identified with electrons from broken cooper pairs. Since the quasiparticles in this case are just electrons, they obey fermi statistics. And they will tunnel across your barrier there to minimize the free energy of the system.

My answer will be brief unfortunately. Here's how I think about it. Electrons in the band that is near the Fermi energy interact through some attractive force. The ground state of this system is the superconducting state, with all those electrons paired up and you have a superconducting condensate which is that "Cooper sea" Now we consider low-energy excitations of this state. Turns out this just comes about by breaking one of this Cooper pairs (by thermal excitation or otherwise). This costs an energy equal to the superconducting gap. So those quasiparticles, which are just the elementary excitations of the ground state (creation operators acting on the vacuum state), can be identified with electrons from broken Cooper pairs. Since the quasiparticles in this case are just electrons, they obey Fermi statistics. And they will tunnel across your barrier there to minimize the free energy of the system.