It doesn't occurr because of an increase in energy - that would be normal conduction.
Let me see if I can explain it a different way. When the doping concentration is very high on both sides of the junction, the depletion region is very thin. In general, carriers are repelled by this barrier since they are not energetic enough to surmount it.
However, as the barrier gets thinner and thinner, quantum mechanics tells us that there is some small but finite probability a carrier is not repelled and instead is found on the other side - and !blink! - it is.
It's the equivalent of throwing a ball against a wall. The probability that it will bounce back is enormous, but after billions of years qm says that one day it will simply pass through and appear on the other side. Try it and see!