Why there is a potential energy barrier here?
This is the simple barrier penetration quantum mechanical picture .
Note that the tunneling does not change the energy state, just the wavefunction, i.e. the probability.
In the image you post, because the electrons are bound within the sample and the tip they cannot go through the air. In the air they have to have an extra energy, a different energy level , than the bound levels in the material. This difference in energy levels is the barrier.
They tunnel through:
Scanning Tunneling Microscope is based on the concept of quantum tunneling. When a conducting tip is brought very near to the surface to be examined, a bias (voltage difference) applied between the two can allow electrons to tunnel through the vacuum between them.
The bias voltage brings the same energy level between the tip and the sample, and so allows the tunneling probability to be high, as in the link above .