I am not very versed on Deep Space Nine but I keep seeing clips about the Founders. They seem to have a home planet on the other side of the wormhole and keep convincing other species to aid them in conquest.
What exactly are the Founders offering all of these species?
Yes, they can shape-change and there is the possibility of powerful people being fake, but there seem to be incentives that I missed. Why are they considered so powerful? Is it all diplomatic power?
The Founders are the covert rulers of a vast space empire. At some point in the distant past they managed to genetically corrupt two species; the Vorta and the Jem'Hadar, using them as tools to take over gargantuan numbers of planets and a chunk of territory several times the size of the Federation. Note that a key part of their control mechanism is that they're quite literally worshipped as living deities.
The Vorta
WEYOUN: And living in fear of the many predators that would hunt us for food. One day, a wounded changeling came stumbling through the forest fleeing from a mob of angry solids.
ODO: Why were they chasing him?
WEYOUN: What does it matter? Solids have always feared and distrusted shape-shifters, you know that. Well, a family of Vorta hid the changeling from his pursuers. And in return for saving his life, the changeling promised the Vorta that one day we would be transformed into powerful beings. That we would become an important part of a great new empire that would stretch across the Galaxy.
ODO: And the changeling kept his word.
WEYOUN: That's right. Imagine, Odo. My people were once little more than apes, and look at us now. Look at what you've done for us.
The Jem'Hadar
"They were some incredibly nasty, conquering subculture on a world of their own, but without all the genetic engineering; they didn't grow up in three days and all that stuff. The Founders got a hold of them and said, 'We'll make you the ultimate killing machines, what do you think?' And they said 'YEAH!' They just volunteered."
The Founders offer the divided regimes on the Alpha side of the wormhole the advantages of alliance with a powerful force, and the threat that the Founders would ally against them.
What makes the Founders so powerful?
The Founders have bred a species of ferocious fighters who are under the founders' complete control through use of a material which the fighters (the Jem'Hadar) require to live (Ketracel White). The Founders also bred a species of cloned diplomats (the Vorta) who worship the Founders as gods. Together, these are formidable advantages.
The Founders also do not fight among themselves, so they are not susceptible to being divided against themselves. Even without the power to shapeshift, and thus subvert, confuse and corrupt their enemies, the Founders would be a powerful foe - and with their transforming powers, they are a threat to any Federation-level society (or even a coalition of Federation-level societies).
It's not necessarily transactional. The Founders more-or-less created the Vorta, and in so doing, built in an instinctive, compulsory loyalty, causing the Vorta to see the Founders as Gods. In creating a powerful warrior race such as the Jem'Hadar, loyalty had to be enforced, hence the dependency on Ketracel-white. Since the Vorta unconditionally worship the Founders and are in control of the supply of Ketracel-white effectively enslaving the Jem'Hadar, the Jem'Hadar are essentially obliged to worship the Founders also. It's a potentially fragile arrangement; the Founders would be defenseless against overthrow and/or destruction at the hands of the Jem'Hadar; they don't because conditioning prevents them from even imagining the idea.
The Founders are powerful because of the military threat from the Jem'Hadar at the command of the Vorta.
I should add that the Founders could also exert influence through infiltration. Although Odo, the female Changeling portrayed by Salome Jens, and the Changeling Laas assumed humanoid form rather imperfectly, others could assume human form so convincingly that they had successfully infiltrated Starfleet Headquarters (DS9: "Homefront"). That had the potential to make the Founders a powerful force, barring countermeasures by an informed adversary.
What makes them powerful is simply good technology. If they didn't have that, they would not have been a problem at all.
What makes them a problem is that they set themselves up as Lawmakers, Police, and Judges. Often in the same individual. Often with no notion of informing others of laws. Often with no notion of not doing ex post facto laws.
The later is where Odo and the Founders differed: Odo was just police. (Though I suspect he would be willing to enforce any law on the books, even if it wasn't normally enforced.)
From my memory:
BEWARE SPOILERS if you did not watch DS9 to the final season:
The power of the Founders wasn't over the Jem'Hadar/Vorta alone. There were many other worlds/races, but they were not plot relevant.
It was said the Founders were some kind of conquerors, because they sent changling "seeds" to many worlds to "grow up there, integrate and observe and to learn" (aka "how to control them") and then were recalled to the "great unity" (or how they called that "lake" of changlings, can't remember).
MORE SPOILERS concerning Odo:
Odo was special, because he lost contact to the Founders due to traversing the wormwhole (by random coincidence, not planned) and being too far away for the recall. Therefore he was glad to reunite, but realized later, how cruel they were (episodes with this "female" changling; I don't have an exact source). He then sided with the federation against the Founders again.
