The 24-year-old joined the nine-times European champions last week in a whirlwind transfer worth about 12 million euros ($14.79 million) after spending his entire career at the Premier League club.
"I gave 100 percent for Liverpool but, when you have been somewhere for 13 years, maybe it comes too naturally," he told Monday's Times newspaper.
"It is the same route to work every morning, the same faces and you are the top goalscorer every season. You are not reaching for that extra yard from yourself and it is not for the want of trying. That is just nature, it is life.
"Now I'm certainly going to have to stretch myself. I am pleased and proud of myself for making that step out of the comfort zone and going for something that is a bit scary, a bit nerve-racking.
"Part of my brain was saying 'just let your career run down, you're safe here, your family is around you, everyone likes you.' But I felt the drive to better myself as a person and as a player."
Owen teams up with England team mates David Beckham and Real's latest British purchase, defender Jonathan Woodgate, in the Spanish capital.
Despite his prolific scoring record, with 23 goals in 60 England matches and 158 in 297 games for Liverpool, he will have to fight stiff competition for a place in the starting lineup.
"Now I have to prove myself at Real and I know that, in Ronaldo, Raul and Fernando Morientes, they have three of the best strikers in the world," said Owen.
"I am different from all of them, which can only be a good thing. I expect I'll have to sit on the bench once or twice, but so will everyone."
"It is a challenge but that is why I came here. I've had this sneaky suspicion all along that it will work out well," added Owen, who said he was disappointed by some of the "defeatist reaction" to his transfer.
"I don't know if it's an English trait to settle for what you've got, but I was taken aback when the first thing some people said was 'how's he going to get into that team?'"
Owen said his decision to leave Liverpool was driven by ambition, after Liverpool failed to match Arsenal and Manchester United as a championship-winning team.
"Money never came into it, just pure ambition. To have stayed would have been like anyone turning down a promotion. If you are going to be a top, top player then you have to win the league.
"I have total respect for Arsenal and Manchester United lads who keep winning everything, but it does fill you with jealousy, envy. When you see their medals, you think 'I could be doing that,'" he said.
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