Anatoly Sobchak (whose daughter Ksenia was Russia's "Paris Hilton" for a long time before she became politically active in the anti-Putin opposition more recently) was the mayor of St. Petersburg in the 1990s. He was an economic reformer and seen as a capable administrator. Putin was in his team and showed himself to be extremely capable and talented. He also, of course, had the right connections to the FSB and other security groups so he was seen as the sort of person who would listen to Russia's power brokers (the army and the secret police have always been the only two sources of real power in Russia, joined briefly by the Communist Party during the Soviet period).
Yeltsin's chief of staff, Pavel Borodin, engaged for months in very delicate negotiations with Putin and his team regarding a transfer of power. Everyone knew Yeltsin was in terrible health and his family was anxious that, if he were to die in office, his successor would immediately arrest the Yeltsin family for corruption and massive looting of state assets. Borodin negotiated an agreement that Yeltsin's entire extended family would get the Hell out of politics in Russia forever and in exchange they wouldn't be touched.
This applied also to the oligarchs of the 1990s. They were told to stay out of politics and they would keep their fortunes. A few ignored this demand - Boris Berezovsky (fled to London, then hanged himself under strange circumstances), Mikhail Khodorkovsky (jailed, then exiled) and a few others who were jailed, killed, fled or ended up caving and reconciling (and paying extra money for the privilege).
However, almost everyone in politics knew that Putin was going to be "cleaning house" and that a strong state was returning. The question was just how strong the state would end up being, given that Russia's historic, geographic, climatic and demographic conditions have always favored a radical centralization of power.
ἡ δὲ κἀκ τριῶν τρυπημάτων ἐργαζομένη ἐνεκάλει τῇ φύσει, δυσφορουμένη, ὅτι δὴ μὴ καὶ τοὺς τιτθοὺς αὐτῇ εὐρύτερον ἢ νῦν εἰσι τρυπώη, ὅπως καὶ ἄλλην ἐνταῦθα μίξιν ἐπιτεχνᾶσθαι δυνατὴ εἴη. – Procopius
Ummaka qinnassa nīk!
*MySmiley*