The following contains spoilers for Obi-Wan Kenobi, Season 1, Parts I and II.
The first two episodes of Star Wars: Obi-Wan Kenobi didn't do anything very unexpected, but they were on point -- giving fans everything they had hoped for. As the series began, Obi-Wan was still wallowing in the grief and guilt associated with Anakin's fall to the dark side and the subsequent decimation of the Jedi Order. Ewan McGregor's portrayal of Obi-Wan's broken state had viewers both impressed with his performance and devastated at the aftermath of Order 66.
However, things quickly became much worse for the Jedi Master. In a change from Legends material, Obi-Wan learned (at the very end of Part II) that Darth Vader survived their fiery, fateful duel in Revenge of the Sith. That was a full decade after he left his former apprentice for dead beside a sea of lava on Mustafar. Here's what happened and how Obi-Wan was able to maintain ignorance for so long.
In Star Wars Legends continuity, Obi-Wan found out about Vader right after Revenge of the Sith. The story was told in the final pages of a novel by James Luceno, called Dark Lord: The Rise of Darth Vader. Obi-Wan was sitting in a cantina in Anchorhead when he saw a report on the HoloNet about a Darth Vader killing a whole band of Jedi on Kashyyyk. The news sent the Jedi master for a loop and nearly gave him a full-on panic attack.
Obi-Wan Kenobi, however, decided to rewrite that story. When the second trailer came out, some fans believed that it showed the moment when Obi-Wan realized that Vader had survived. Others heartily disagreed with that theory, but it ended up being correct because the moment was in Part II.
Over the first two episodes, the ambitious Third Sister (Reva) set up a plot to lure Obi-Wan out of hiding. She had a band of smugglers capture Princess Leia from Alderaan. Faced with the news, Bail Organa pleaded with Obi-Wan to rescue his daughter. It took some persuading, but Obi-Wan relented and traced the kidnappers to Daiyu. While there, Reva pursued them directly, and that's when she dropped the news. As a way to intimidate Obi-Wan, she announced that Vader (whom she identified as Anakin Skywalker) was alive. As expected, the news threw Obi-Wan for a loop, and Reva might have gotten the better of him. However, the Grand Inquisitor showed up and tried to take credit for Obi-Wan's capture. Their ensuing argument and stabbing allowed Obi-Wan the opportunity to escape.
It might seem odd that Obi-Wan could have made it 10 years without hearing about Vader, but it makes sense for two reasons. First, all indication is that the Jedi Master cut himself off from the Force for those 10 years. That kept his presence hidden from the Inquisitors, but it also meant that he couldn't have felt Anakin's presence in the Force. Supplementing that is the fact that Obi-Wan led a solitary life on Tatooine. In Legends, he found out the news in a cantina, but it's perfectly viable that he wouldn’t have seen anything about Vader. The less he knew about the galaxy, the better, and it's clear that he was taking his self-exile pretty seriously.
Of course, his exile fell apart when he finally decided to rescue Leia. And now, he's going to have to deal with Darth Vader during the rest of the series. They are supposed to have the "rematch of the century," so it will definitely be a fight worth seeing.
To see how Obi-Wan responds to Vader's survival, watch Star Wars: Obi-Wan Kenobi. The first two episodes are streaming now on Disney+.