It was a bit of a shock for Yeltsin to announce his resignation when he did, but it was not totally unexpected. While he was seen (in the West at least) as the one who helped bring down the Soviet Union, he had had a troubled presidency and was quite unpopular domestically by this point. Arguably the cause of this was the ill-fated decision to launch the Second Chechen War in August 1999, which was meant to resolve the issues the First War (which had nominally ended in 1996) had left behind. Notably the newly-named prime minister, Vladimir Putin, was given control of the war, and oversaw its operations. But to get to this point I'll quickly go over how Putin became prime minister, and subsequently was named president.

Putin was a KGB officer, who spent several years in Dresden, East Germany, and was there the night the Berlin Wall fell in 1989. This has been cited, by Putin himself and by those who have written about him, as having a major impact on his life: he's noted that there were crowds of Germans outside their offices and when he made a call to a superior on how to deal with it, was pretty much told they were on their own. This breakdown of authority has thus been cited as a major event in Putin's life.

He returned to Leningrad in 1990 (which was to shortly go back to its original name, Saint Petersburg), and ended up in the office of the mayor, Anatoli Sobchak. Putin served as an adviser on international trade, and developed a close relationship with Sobchak, which likely helped him keep his position (there are reports of questionable conduct by Putin in this time, but being Russia in the early 1990s, that was to be expected).

Sobchak lost his seat in 1996 so Putin moved to Moscow, and took up a role within the federal government. From there he quickly moved up the ranks, and by 1998 was the head of the Federal Security Service (known by its Russian acronym, FSB), the successor to the KGB. From there he became acting Prime Minister in August 1999; while this seems like a big step, note that at this time Yeltsin was going through prime ministers in a crazy fast manner: Putin was the fourth person to hold the title in 16 months. But Putin managed to hold on, and was the one left when Yeltsin finally resigned.

Regarding Yeltsin, he had not had a great presidency. The first democratically elected leader in Russian history, he did not have the support of the Duma (Russian parliament), and famously shelled the parliamentary building in 1993 after they refused to follow his orders (he had initially been given expanded powers to help transition Russia into a market economy). Subsequent elections in 1995 saw the Communist Party win a huge plurality of seats, and with the continued economic woes (the 1997 economic crisis badly damaged the Russian economy), Yeltsin lost any popular support. Factoring in his decision to launch the wars with Chechnya, and it is not surprising that he eventually resigned.

Regarding some literature, I really found The New Tsar: The Rise and Reign of Vladimir Putin (2015) by Steven Lee Myers to be balanced, informative, and thorough. He covers the life of Putin, as well as the 1990s political and economic situation in Russia, while largely keeping to facts and not giving opinion (which some other books on the topic do). It would be a good start to getting more in depth on the subject, and I can't recall but it may even have a bibliography at the end for some further research.