In quantum mechanics, you don't know the exact position of either electron. So when computing the energy of an orbital, we really compute an average of the energy over possible positions of the electrons, weighted by the probability of each configuration.
The ultimate origin of the exchange energy for electrons is the Pauli exclusion principle, which states that two electrons cannot be in the same position at the same time. This in turn comes from a fact in quantum mechanics, that is difficult to explain in simple terms, that the quantum mechanical wavefunction describing the system must change sign when we interchange two electrons.
While the origin of the Pauli exclusion principle is admittedly mysterious, it has a clear effect on the average over electron positions. The electrons have a repulsive energy
\begin{equation}
U = \frac{ke^2}{r}
\end{equation}
where $r$ is the distance between the electrons. If we were to ignore the Pauli exclusion principle, the average would include configurations where the electrons were very close and had a large repulsive energy. The Pauli exclusion principle means that configurations where the electrons touch are not allowed, and since the wavefunction is smooth is also means the probability for the electrons to be "close" is reduced relative to the case where we ignore the exclusion principle. So the average energy is smaller when accounting for the exclusion principle, because less weight is given to configurations with a large repulsive energy between the electrons. (Note that a full calculation will involve other terms in the energy, and the antisymmetry of the wavefunction is crucial to understand what is going on in full detail, but here I am just focused on getting across the main point in a simple way).
Even though the Pauli exclusion principle is not really a contribution to the energy, the difference between the average energy you compute accounting for the Pauli exclusion principle, and the energy you compute without accounting for it, is called the exchange energy. It encodes the net effect of the Pauli exclusion principle as if it were a contribution to the energy, even though it is not, strictly speaking. The exchange energy is negative, because the actual energy is lower than what you get ignoring the exclusion principle.
Note that the exchange energy is negative for electrons because they are identical fermions. For non-identical particles, the exchange energy is zero. For identical bosons, the exchange energy is positive, because the symmetry (instead of antisymmetry) of the boson wavefunction assigns additional probability to configurations where the bosons are near each other, compared to the non-identical case.
