I am not able to figure out how the compressed position of a pogo contact is retained. Unless you apply a significant force external (against the spring,) wouldn't it try to release the plunger back?
Its compressed position is retained by exerting an opposing force on whatever is making contact with the pin - you can't just rest the contact point on top of a pin and expect things to work.
Regarding "significant force" - one pin alone is often negligible, but the force increases as you add more.
Pogo pins are used for all sorts of connections - from batteries and other quick connect systems through to test jigs.
Without a movable "head", in any situation where the two halves don't have perfect / infinitesimal tolerances (and thus are extremely expensive to produce) you'll find that some points don't make contact due to high or low points. Allowing the head to move means that the tolerances can grow (often significantly) with a good connection still being made. Of course, you can't just let the head move freely, as it'll fall due to gravity, so a spring is used to keep it "up", but it allows movement to absorb these differences.
The spring also provides force between the contact points - it's actually possible to have two conductors physically touching but not making a good electrical connection (or potentially any connection at all - see Wetting current). Gravity alone is fairly weak, and sometimes can't penetrate the oxidie layer by itself. You can often see this effect if you're very gentle with multimeter / scope probes - we don't just rest probes on the test point, we apply some pressure.
To summarise, compression is key for two reasons - compliance and connectivity. Look at test jigs, and you'll see that the device under test is actively "pushed" into the bed of nails.
See Adafruit's "How to Build a Test Jig" writeup, and the "Toggle Clamp" they carry... what a good video!
With some test jigs that have a larger pin count, you actually need to be very careful about applying the opposing force - if you just push the corners down, the PCB may flex, and this can be problematic for things like solder joints. The force from a cluster of pogo pins can also add up significantly, exacerbating the situation.
edit:
But I can appreciate this logic only if pogo pins are supposed to be connected to test points.
How is it useful for a connector and socket arrangement? I see manufacturers claiming it being useful in many electronic appliances. If there is a socket and pin connection, why would anyone prefer a retractable 'soft' pin over a solid pin?
I'm confused - you've linked to a video that outlines many advantages and usage scenarios.
It's not so much that the pins are "retractable" (which would suggest they can be stowed or put away), but rather that they will A) exert their own force, and B) conform to a non-uniform or ill-defined mating position.
The MagSafe connector is one very well known example which even has the "pogo pin" appearance, but if you have a smart watch, take a look at the charging connection.
Also, remember that spring-loaded contacts are used extensively all over the place - in fact most connectors aren't just "a solid pin" as you suggest, but have at least one spring-loaded component in the connection. Even mostly-permanent internal connections regularly use sprung contacts (e.g: between a phone's mainboard and an LDS antenna, speaker, etc...). The only good example of non-sprung connections I can think of are where either a nut and bolt are used (e.g: cat battery terminals), or press-fit connectors or teminals (i.e: pushing a pin through a hole that is notionally "too small")
