Methylene "anti" Jahn-Teller effect
The Jahn-Teller effect is not relevant here. The triplet state is observed in many molecules to be of lower energy than the singlet. The Pauli exclusion principle, makes the states symmetric or anti-symmetric to the exchanging electrons.
A triplet state has unpaired spins while in a singlet they are paired. The triplet is lower in energy than the corresponding singlet. One reason for this is that the electrons try to ‘avoid’ one another in the triplet; (see Salem 'Electrons in Chemical Reactions', publ Wiley 1982 for a detailed discussion). This means that the spatial part of the triplet wavefunction is asymmetric to exchange of the electrons;
$$\varphi_T = \psi_1(1)\psi_2(2) − \psi_1(2)\psi_2(1) $$
and the singlet spatial part is symmetric which means that the electrons tend to crowd together raising the energy
$$\varphi_S = \psi_1(1)\psi_2(2) + \psi_1(2)\psi_2(1) $$
The numbers in brackets refer to the electrons’ coordinate. As the total wavefunction is anti-symmetric to exchange of electrons the corresponding spin parts have the opposite symmetry to the spatial parts.
If the spins are on two separate atoms then the singlet and triplet energies can be very close to one another and the order can be changes by external magnetic fields. There is more detail on this effect in this answer How does spin flipping of triplet carbenes occur?