The Icelandic band have already reached number six on the US Billboard 200 with their first record My Head is an Animal and are starting to adjust to crowds knowing their tracks.
"It's been pretty strange. When I'm at concerts I'm always like, 'Wow, how does it feel to have loads of people know your songs?' Now I'm getting to know it," Brynjar Leifsson told Digital Spy at Reading Festival.
Singer Nanna Bryndís Hilmarsdóttir explained that she would like the success of their album to continue in the UK after its release on Monday.
"We'd love to keep that going, but whatever happens happens. If people like it, it's great," she said.
Leifsson added: "I wouldn't hate it being a success in the UK! If it does well, it means we get to come here more often."
Following the breakout success of their single 'Little Talks', the track has charted all over the world and launched Of Monsters and Men to international status.
"We're still kind of realising that," Leifsson said. "Because things have been happening so fast for us - we've only been touring now since March - it's hard to wrap your head around it all.
"We've been playing America and Australia a lot and now we're in Europe, so it's just sinking in that this is actually full-time."
Of Monsters and Men's debut album My Head is an Animal is available to buy now.
Watch the music video for Of Monsters and Men's single 'Little Talks' below: