Jumping sewer lid - WHY?

This phenomenon has been well-researched. If the underground pipe is almost full with rapidly-flowing water the waves in the pipe will cause the pipe to be completely filled in some places. The waves are irregular and the air in between the wave crests gets compressed - when it passes the manhole access the air is vented. 1000Pa is actually a very small amount of pressure - about 0.5% of what is in a car tire, and can easily be created by irregular wave functions in a pipe.

Increase the pressure a bit and even large manhole lids can rocket out of their housing and become a spectacularly dangerous projectile. In areas where this phenomenon is likely you will find the covers bolted down. Not locked, just some easily-removed large bolts to keep it where it should be.

I will vote for

Water filling the sewer so quickly, that it made the air pressure this strong? Would this really be the easiest way for the air to leave the sewer?

In the two videos John Rennie linked to it is evident that the air is coming out with pressure aerating the water which is high around the manhole. In your video not enough water is seen around the manhole.

Manholes are usually over wells where several sewer lines join to go down a common exit line. With a severe enough storm all entry routes to the well may be full so they act like pistons, pushing the air ahead of them raising its pressure and part of it is enough to raise the lid, release a bit of pressure and raise it again. If the storm were severe enough filling up the well, the cover would have been lifted by the water. This shows how high the water pressure can get, depending on the design,

The answers in the comments may be possible, but unlikely. Air would be able to escape through other places such as storm drains/gutters. Its seems more likely to me that the lid is not being pushed up by high pressure underneath so much as it is being pulled by low pressure above the hole. It has to do with something called Bernoulli's Principle.

You can read the article to get the full idea of the physics behind this, but the basic idea is that a moving fluid has a lower pressure than a fluid with the same properties and conditions except for being at rest. We use Bernoulli's principle all the time in everyday life without thinking about how it works. For instance, an airplane's wing is designed with a specific shape so that air moving over the top of it moves faster than the air below it, causing a pressure differential that creates lift on the wings.

Thinking about it this way, it makes sense that during a storm, wind blowing over the top of a manhole cover could cause a low pressure area, while the still air underneath the cover would still be at normal pressure. With fast enough winds, it could easily create enough of a pressure differential to lift the cover, making it "jump". Of course, as soon as the lid is lifted, air rushes out and the pressure equalizes, so the lid goes back down.