There is a questline that leads to it. The marketing was pretty clear about it. The raid stuff is there.But it isn’t emphasized by the game to go to the DC anymore than opening up Titan did from the EDZ. There is still so much to do elsewhere with flashing icons on the director.

How can one level up quickly here?

You get Powerful Drops in the Dreaming City from:

Your first Blind Well Tier 4 completion
Petra's Blind Well Bounty (2 Tier 4 completions finishes it).
Petra's Ascendant Challenge Bounty (find and complete the weekly ascendant challenge, very fun activities)
Completing both of the above bounties gets an extra powerful drop.
The weekly 8 dailys bounty.
The weekly story.
Shattered Throne encounters, if Max Curse Week.

It is a little weird that in game The Dreaming City isn’t more clearly pushed as the core Endgame for Forsaken. It is still on of the most efficient places for time-sensitive power growth. It’s also the most engaging content to repeat week-after-week due to being inherently different eqch week.

But the game doesn’t do the best job selling that.

Yeah, I've been in the Dreaming City maybe a dozen times. Other than boring weekly missions, a once a month Dungeon, and a raid that I have zero interest in, I don't really see what makes it any different than any other place in the game. Certainly nothing tells me that it is an efficient place to level up (and honestly, your post is news to me).
The Escalation Protocol event in the Dreaming City doesn't seem to drop powerful gear, and it causes me eye strain to be there, so I haven't even unlocked the majority of my extra Supers.

To me, it doesn't even look remotely as interesting as people seem to hype it up around here (then again, I may have been spoiled by God of War as far as fantasy settings go)...

How can one level up quickly here?

Each week, you have the story mission (1 of 3 on a weekly rotation), the ascendant challenge (1 of 6 on a weekly rotation), and Petra’s weekly challenge that all reward powerful drops. Petra’s challenge involves doing some Blind Well to obtain an offering for the Oracle, as well as completing 8 daily bounties.

In addition to the powerful drops, many of the activities will provide regular legendary drops. Blind Well, the Oracle offering, many of the bounties, chests, powerful enemies, public events, ascendant bounties, etc. This isn’t anything special on its own, but the Dreaming City is a bit more of a full-featured loot ecosystem than the other destinations. It has a full set of unique and good weapons, plus the Dreaming City armor is one of the few sets in the game that can roll with enhansed perks.

So, on a single character, my average weekly trip to the Dreaming City will yield 10-15 legendary drops (not including Prime engrams), 3 of which will be Powerful drops. With so many drops from the DC gear set, it’s faster to build up a collection of armor and weapons that have the perks you like, and then use the powerful drops to keep your gear at max light without needing to spend Cores. Basically, it makes it easier to brute-force your way past all the crappy RNG barriers than in any other destination.

On top of all this, there’s the fact that many of the Black Armory bounties can also be completed in the Dreaming City. So that’s a bonus.

This is all without touching the ascendant chests, hidden activities, Shattered Throne (3 more powerful drops per character when its available), or the raid. Mechanically, I don’t actually think of the Raid as part of the Dreaming City, because it has its own self-contained loot pool. But if you do get into the raid, it has another great set of weapons that are loads of fun to use.

As said above by others about drops, that covers most of the big stuff. So in an hour you can hit the story mission, Ascenent Challenge, and Blind Well done. Blind Well with 2 Tier 4 completions gets Petra’s Bounty and the completion there. That is 5 Powerful drops. In contrast on Day 1 of a Flashpoint you can do a Heroic Adventure and finish the Flashpoint for 2 Powerful drops, in a comparable timeframe. A second visit later for a second Heroic Adventure is 3 Powerful drops. With a second visit to DC you can complete the 8 Daily Bounties for Petra and get a 6th drop.

You can squeeze a lot out of it compared to other activities. Not that you should limit yourself if you are just after powerful drops, but there are lots of good reasons to do stuff in the DC if you are pressed for time and not specifically trying to pursue other activities.