The versatile 29-year-old Ghana midfielder, who played under Real coach Jose Mourinho at the London club and refers to the Portuguese as "daddy", flew into the Spanish capital and passed his medical on Saturday.
"I have had my injuries but I am in good form now and very strong," a beaming Essien, who will wear the number 15 shirt for the Spanish champions, told reporters.
"If I wasn't ready Real Madrid would not have brought me here," he added.
"Daddy (Mourinho) knows me very well, he brought me to Chelsea, and he knows I can play in any position he wants - except goalkeeper."
Essien's 2011-12 campaign was ruined by a knee injury he sustained in pre-season, the last of three serious setbacks in recent years, and he struggled to win a regular place in the Chelsea starting 11 on his return.
Mourinho has brought Essien into the Real squad to replace France midfielder Lassana Diarra, who sealed a move to wealthy Russian club Anzhi Makhachkala on Friday pending a medical.
Essien, who like Diarra can also play at right back, said the call from Mourinho had come as a surprise and the negotiations had been concluded rapidly.
"Everything went very fast and I was in my hotel room and I had a call from my daddy and he said he would be very happy to have me at the club," he said.
"I never thought I would end up here. I was focused on Chelsea and I didn't know that Real Madrid would make a bid for me.
"When Mourinho calls it's more than enough. All I want to do is play football and when he rang me I said yes immediately.
"When I arrived he was at the airport. We were really happy to see each other and I am really looking forward to starting playing and enjoying the football."
Essien signed his one-year loan contract in the presence of Real president Florentino Perez and was given a watch, a Real shirt, a model of the stadium and a fountain pen.
After taking only one point from their opening two matches, Real are looking to get their bid for a second consecutive La Liga title back on track when they host Granada on Sunday.
Mourinho's team claimed their first piece of silverware of the season by beating arch rivals Barcelona on away goals to clinch the Spanish Super Cup on Wednesday.