Yeltsin was impressed by Putin’s loyalty to his political mentor Anatoly Sobchak, former mayor of Saint Petersburg.
Sobchak was persecuted by some of Yeltsin’s underlings because for a time he was perceived as a Yeltsin’s competitor (Sobchak was well known and respected in the country and abroad). Sobchak was on the verge of arrest for minor corruption. He was also seriously ill. An arrest meant death for him.
So, Putin took a vacation from working in the Kremlin and arranged for Sobchak an escape to France. It was a whole international secret operation, practically a caper.
Because Sobchak was previously a personal friend, Yeltsin wasn’t enjoying his persecution. He was glad that Sobchak just left the country for good. And President was impressed by Putin’s willingness to risk his career and freedom in order to save his former boss, who at the time was just a helpless old man. Yeltsin saw that as a guarantee of his own safe future. Such loyalty was too rare not to use it.