Warning: This article contains spoilers for Obi-Wan Kenobi episode 3.
Darth Vader's Inquisitors, Reva, discovered the past connection between Obi-Wan and the Organas when studying the archives, and came up with the idea of having Leia kidnapped to draw him to the planet Daiyu.
Reva's gambit was a success, but Obi-Wan and Leia managed to escape Daiyu nonetheless. They stole aboard an automated cargo ship, one headed to the mining planets of the Mid Rim; the Empire was initially unable to identify precisely which ship they'd stolen aboard, but they were able to send Imperial Probe Droids out to potential targets. It didn't take long for Obi-Wan to be identified on the planet Mapuzo, a world he recognized as once fertile grasslands now turned into an arid wasteland. And yet, even here Obi-Wan found hope - a secret network that had been ferrying Jedi and Force-sensitives to safety. Ironically, it's entire possible this was set up by Leia's adopted father Bail Organa himself.
Unfortunately, Obi-Wan's presence on Mapuzo led to the Empire arriving on the world in force. It didn't take long before that part of the network was compromised by Darth Vader's Inquisitors. Meanwhile, in a shocking twist, Obi-Wan Kenobi episode 3 saw Vader himself arrive on the planet to lead the search for Obi-Wan - and the barren world of Mapuzo became site of the show's first duel between Obi-Wan and Darth Vader.
Why Darth Vader Is So Much More Powerful Than Obi-Wan
Obi-Wan soon realizes he is completely outclassed by Darth Vader. Obi-Wan is a master of Soresu, the most defensive of the seven lightsaber forms - and one that requires a close connection with the Force. The Jedi Master hasn't been training with his lightsaber for almost a decade, his connection to the Force is weakened, and Darth Vader immediately places him at a disadvantage. The Dark Lord of the Sith opens the duel by torturing innocents, ensuring Mapuzo became shrouded with pain and fear in the Force - things that cut a Jedi off from the light. Darth Vader's more aggressive Shien allows him to hammer at Obi-Wan; Vader has learned to put all the weight of his armor behind his blows, and at several points he almost knocks Obi-Wan's lightsaber out of his hand.
Darth Vader's duel with Obi-Wan then takes a shocking turn, as the Sith Lord uses telekinesis against his former Master. It's actually quite rare for battles between Jedi and Sith to be resolved in such a demonstration of power, with the old Expanded Universe explaining duelists create shields of the Force around themselves. By this reading, Darth Vader's actions here are a way of humiliating Obi-Wan, revealing his defenses are nothing against the power of the dark side he embodies. Darth Vader then proceeds to torture Obi-Wan in revenge for Mustafar.
Why Darth Vader Let Obi-Wan Go
Fortunately, Obi-Wan has help. In truth, though, Obi-Wan Kenobi episode 3 makes it quite clear Darth Vader allows his target to escape; the Dark Lord of the Sith has nothing to fear from a single sniper and some flames. Darth Vader is clearly toying with Obi-Wan, drawing out the chase so that his enemy feels a taste of hope, only to have it snatched away again. Sith are encouraged to indulge their anger and hatred, and there is nobody Darth Vader hates more than Obi-Wan.
It's also possible Darth Vader has intuited that he has stumbled upon a network of some kind; if this group is helping one Jedi, they may well have assisted another. The Emperor has charged Darth Vader with responsibility for hunting down all Jedi, and his pursuit of Obi-Wan has the potential to expose this entire network and even uncover other Jedi. There would be a twisted irony if Obi-Wan's return led to the deaths of more Jedi, and Vader would find that idea particularly appealing.
We Already Know Who Wins Kenobi's Real Obi-Wan vs Vader Duel
The first encounter between Darth Vader and Obi-Wan has not gone well for the Jedi Master. But Lucasfilm has already confirmed there will be a rematch; this is the first of two duels that will take place in the Obi-Wan Kenobi Disney+ TV series. Obi-Wan must triumph in the second encounter, simply because when they next meet in the first Star Wars film, Darth Vader will brag about his powers have increased still further, and he is the master now. But it's likely even this won't be a straightforward victory for the Jedi; Obi-Wan will probably fake his own death, simply because anything else would leave Leia in danger. The Empire now know they can draw him out with Leia.
How Did Reva Find Leia So Quickly (& What Happens Next)
That will happen sooner rather than later, because Obi-Wan Kenobi episode 3 ends with the young Leia in the hands of Reva. The Inquisitor discovered the network, and presumably deduced where the secret age was headed - thus getting ahead of Leia herself. This means Leia will once again become bait in a trap for Obi-Wan, but this time the trap will be darker still; if he is to get Leia back, he will need to confront Darth Vader and the Inquisitors themselves.
This is the moment of greatest danger for the entire galaxy. The Inquisitors are used to looking for children who are strong in the Force, and it is entirely possible Reva will realize Leia's potential. Worse still, Obi-Wan was reminded of Leia's mother Pé when he looked at her, and Darth Vader could find her just as familiar. Obi-Wan will need to get back on his feet and rescue Leia quickly, with the fate of the galaxy at stake. Obi-Wan Kenobi episode 3 really does set up a dramatic future for the show.
New episodes of Obi-Wan Kenobi release on Wednesdays on Disney+.