The Republic of Ireland midfielder joined the Toffees from Wigan for an initial £13m fee last week and is prepared to bide his time before making his mark.
He said: "It is a price tag and one I need to deal with. I am looking forward to trying to prove why I am here.
"It is not going to be easy, we have got a lot of competition for places. I am going to give it my best shot and hopefully I can do well."
McCarthy admitted that the opportunity to renew acquaintances with his former Wigan manager Martinez was a key factor ahead of the switch to Merseyside.
"He is a very good manager," he said. "You can see he likes to play football. It is always pleasing on the eye for people watching.
"It is great to be here with him and hopefully I can stamp my authority on the team and get as many starts under my belt as possible.
"But I have got real competition at the club. There are a lot of big names here. If I need to bide my time, I will bide my time."
Before his move to Wigan in 2009, McCarthy was handed a trial at Liverpool while he was still with Hamilton Academical.
He said: "I went down to Liverpool for a week's trial. It wasn't to be, I was too young at the time. I didn't want to move away from home.
"To be honest, the way things have worked out, I am happy.
"It was very hectic. At times it goes through your head that it might not happen.
"As time went on, it got to 10.30pm at night and I was thinking to myself, 'Is it going to happen or not?'
"I'm just delighted to be at such a great club. It is a great place to be. Since I heard they were interested I wanted to come here."