The jump is most likely due to several factors.
The main one would probably be similar to what caused the storm in the first place, a high pressure low pressure gradient.
Lifting such a cover would require at least Δp=995 Pa pressure difference between sidewalk level and the sewer.
Someone already derived 995 pascals would be enough to lift a manhole of 50kg and 80cm diameter. Relying on Bernoulli alone at atmospheric pressure would require 41m/s winds.
We also need to account for the fact that the storm reduces the pressure above the manhole.
Air is compressible and water is not at the pressures we're talking about. As the air gets pushed around through the pipes, it is compressed a bit. It will try to escape when it reaches an opening, which could be a manhole or grate.
If the manhole is above a junction, then there are also eddies which further change the pressure gradient. Even without a junction, turbulent flow would create eddies. Normally there is lower pressure above an eddy. The eddy below the manhole is in a closed system so it would just allow buildup of more air below the manhole.
All of this could result in a larger pressure than normal atmospheric pressure under the manhole. Storm drains are generally not connected to the sewer system so during heavy rainfall, the inlets are inundated, trapping even more air with the water.
Compressed air tries to expand when the manhole finally lifts, resulting in the spray of water.
If the air is really trapped and being compressed, someone can test this by just standing on top of the manhole for a bit and see if it lifts higher or sprays more water. Although that is probably very dangerous...