The 29-year-old centre-back enjoyed a superb season with the Blues, and Europe's elite quickly started showing interest with Rudiger's contract coming to an end this summer.
Real Madrid eventually won the battle for his signature, with a four-year contract agreed with the European champions, and the German international confirmed that he turned down offers from Barcelona in favour of their Spanish rivals.
"Yes, there was interest from Barcelona," said Rudiger, "But I told my brother it was Real Madrid or nothing.
"What I can promise is, I will give everything for the club and I want to win as many trophies as possible.
"You come to a big, big, massive club - that's a huge step for myself. But I try to enjoy the pressure.
"I see myself as fortunate to be here, to do what I love.
"I was in the trophy cabinet of Real Madrid and there the pressure starts. This is something that you cannot run away from. You have to try to enjoy it."
Rudiger went on to suggest that the deciding factor in his decision was a conversation between himself and Madrid manager Carlo Ancelotti.
Rudiger will also join up with former Chelsea teammates Eden Hazard and Thibaut Courtois at the Santigao Bernabeu, while compatriot Toni Kroos is another familiar face.
"The most important moment was to speak with Mr Ancelotti and then I made also made my choice that I want to play for this club and under him," added Rudiger.
"The conversation was that he wants me, he trusts in my abilities and that I can be very helpful for him and to the team.
"At my age, where I am right now, this is enough. And, at the end of the day, it's Mr Ancelotti.
"You know in Madrid, you will have competition and in general nothing is guaranteed so you have to fight for your place and that is why I am here, I am ready to do that."