Murray became the first British man to lift the trophy since Fred Perry won the last of three titles at the All England club 77 years ago when he saw off world number one Novak Djokovic in a tense 6-4 7-5 6-4 victory on Sunday.
The 26-year-old, who also beat Djokovic to the US Open title last year and defeated Roger Federer to claim Olympic gold after losing to the Swiss in the Wimbledon final 12 months ago, believes it will take some time for his latest achievement to sink in.
"No-one could really believe it and I was the same," Murray told BBC Five Live.
"You don't want to go to sleep in case you wake up and it didn't actually happen. I was just messaging my friends and laying in bed. It was tough to get to sleep last night [Sunday].
"I'm sure I will see some of the newspapers around. I've seen some of the back pages and front pages of the newspapers this morning [Monday].
"I know I won Wimbledon yesterday [Sunday] but what it actually means - I think that will take longer than 24 hours to sink in and understand it."
Murray is planning on spending a few days with his family and friends to celebrate his first Wimbledon title before taking a short break and then returning to his aim of winning all four grand slam events.
"All of my family, no-one could really believe it. It was just an amazing day yesterday [Sunday]," Murray said.
"I need to try and improve and use this hopefully as a springboard to try and get better. I may never win another slam, I don't know, but I'm going to try as hard as I can and keep working hard and not worry about all of the other stuff that comes along with winning Wimbledon, and after a few days I will enjoy this and get back to work.
"I'm going to try and take a bit of a holiday and go away after the next three or four days. I will see all my family and go out for dinner and see friends and then head off for a week or so."
However, while Murray wants to continue his grand slam-winning success, he admits he is not focused on knocking Djokovic off top spot in the ATP world rankings.
Murray remained second in the men's rankings, behind the Serbian, despite his historic win in Sunday's final.
"It's tough," Murray said when asked if becoming world number one is his ultimate aim.
"You have to be so consistent throughout the whole year. Right now I hold two slams, the Olympic gold and the final of another slam and I'm still nowhere near number one in the world.
"The goal for me is to try and win the grand slams, win those tournaments and not worry too much about the ranking."
Murray added: "When you're on the court you feel what is going on inside the stadium, on Centre Court, but you can't feel everything else that is going on outside.
"But after the match you see some of the pictures of the hill and people watching back in Dunblane and the sports clubs, people at the Tower of London. I don't know how many people watched yesterday [Sunday] on the TV - there will have been hundreds of millions across the world and that's not really something you can grasp. That's a strange feeling.
"When I was sat downstairs on my own when I was waiting to do drug testing, that's when it all hit me. I just got like so tired. I felt like I hit a wall and that's when it felt like it was all starting to sink in, all of the emotions and what I had just done.
"Last year [in the final defeat to Federer] was a tough, tough loss for me, the hardest of my career and the stuff that followed a few days afterwards was not ideal so I'm glad I managed to finally do it yesterday [Sunday], and I can hopefully play without as much pressure and expectation for the rest of my career."