Sometimes, bees will split from a hive, and a bunch of them (including a new queen bee) will swarm, and move to build a new hive.
As you can see, there is an awful lot of bees huddled together around the new queen, they aren't doing much, just clustered the tree branch.
Exactly how dangerous is a group of swarming bees?
A swarm (used in the technical sense to mean that clump you show in your question) is about the least dangerous formation in which you can encounter bees. Their bellies are full (they eat before they swarm) they have no hive to defend, and you are basically not of interest to them. Beekeepers literally gather swarms into buckets to put them into new hives. (Keep in mind, though, that they are familiar with bee behavior and can evaluate the "mood" of a swarm better than you can, so don't get too close.)
As Wikipedia says:
A swarm of bees sometimes frightens people, though the bees are usually not aggressive at this stage of their life cycle. This is principally due to the swarming bees' lack of brood (developing bees) to defend and their interest in finding a new nesting location for their queen. This does not mean that bees from a swarm will not attack if they perceive a threat; however, most bees only attack in response to intrusions against their colony. Additionally, bees seldom swarm except when the position of the sun is direct and impressive. Swarm clusters, hanging off of a tree branch, will move on and find a suitable nesting location in a day or two. Encountering a bee swarm for the first time can be alarming. Bees tend to swarm near their hives or honeycombs, so if a swarm is visible then a nest is nearby. Swarms are usually not aggressive unless provoked, so it is important to keep a good distance from swarms in order to avoid provoking them.
Look at them, take a picture, leave them alone, you saw a cool thing.
And yes, I have met a swarm, one actually set up on a warm sunny day in some old beekeeping equipment (including hives with not-quite-enough-frames) of my father's that we were keeping in a shed. There was a lot of crazy around all of that, but nobody got stung.
Unfortunately, for us as normal, everyday people there is no really good way to tell how dangerous a swarm is.
As Kate Gregory, has mentioned, usually a swarm is the most docile a bunch of bees as there ever was. But there is one fact that trumps all that: we people are stupid.
With normal bees and normal people it would take something like 1,100 bee stings to kill. Granted, it would suck long before that, but for normal bees a normal adult can withstand 10 stings per pound of person. Of course, if you're allergic it could take a lot less.
Even "killer bees" or "Africanized bees" still take that many stings. The "killer" bees are just far more likely to attack, and attack as a larger swarm then "normal" bees.
So, angering the swarm of bees is a bad idea, and you should just stay clear. You are far more likely to be hurt "acting a fool" around the bees than to be actually hurt by the bees.
The swarm itself is not very dangerous, but like many things in nature, not respecting the situation could get you into some real trouble. Your best bet is to remember what our mothers all told us, "If you don't mess with them, they won't mess with you" and move on. If you don't want the bees there then call a beekeeper or pest control depending on your own personal preference.
https://www.ars.usda.gov/pacific-west-area/tucson-az/honey-bee-research/docs/bee-stings-safety/ is a good read
