I do not understand how the Artesian aquifer is causing the water in the flowing well to reach a level higher than the ground surface? And what role does the piezometric surface play in it?
The red line or piezometric line is the level to which the water wants to rise - if it were allowed to reach hydrostatic equilibrium. Artesian conditions are anywhere where a confined aquifer sits below the hydraulic head level (the level to which the water wants to rise). In this case the water is confined and cannot reach the water table even though it wants to - this means it is not at hydrostatic equilibrium. This aquifer is described as having a confining pressure or hydrostatic head. If you put a well into anywhere along this confined aquifer, the water will rise out of the aquifer to the level of red line.
I think in your case its a bit confusing as the piezometric line sitting above the whole of the confined aquifer suggests the whole area is artesian but the 'flowing' conditions are only where the piezometric line sits above the actual surface, meaning the water will actually rise above the surface. Another example is that if you were to dig a well, to the left or right of the highlighted area, the water would rise to that level but would still be below the surface so conditions wouldn't be described as 'flowing'.