The Brazilian Under-21 midfielder completed his permanent transfer to the European champions from Cruzeiro on Monday.
Silva, 21, has signed a contract with Real until the end of June 2020 and was unveiled to the media on Monday at a press conference at the Santiago Bernabeu stadium.
Silva said: “I’m delighted to be here. This is a dream come true.
“I arrive to Real Madrid looking to learn and help. I’m ready to wear this jersey and to honour it.
“This is the best club in the world and I know I’m going to be very happy here.”
Silva, who won the Brazilian league title with Cruzeiro in 2014 and was elected as the best defensive midfield player in the tournament, was a reported target of other European clubs, including Arsenal.
However, the South American was keen on a move to the Spanish capital.
“I’ve always liked Real Madrid, since I was young,” he said. “When I play video games I always choose Real.”
Silva had been linked to Real since August with the Madrid outfit looking for a replacement for Xabi Alonso, who joined Bayern Munich last summer.
“We have signed one of the best midfielders in Brazil,” Real Madrid president Florentino Perez said during Silva’s unveiling. “It’s through our search for talent that we have acquired Silva.”
Perez then looked at Real’s latest signing and said: “You arrive at only 21, having won the league in your country.
“I only ask of you to give your best and show passion for this club.
“You have arrived at a very important stage of the season.
“To wear this jersey is an honour but also a responsibility that only few get to have.”
Silva is Real’s second reinforcement during the winter transfer market following the capture of Norwegian teenage sensation Martin Odegaard from Stromsgodset.
Real are one point clear of Barcelona at the top of La Liga, while they are also alive in the Champions League.
Carlo Ancelotti’s side take on German outfit Schalke 04 in the last 16 of Europe’s most prestigious club competition next month.