The 18-year-old penned a three-year contract with the Reds, who aim to make up for their disappointing displays and results last season in which they finished seventh in the league.
"It's a great feeling to sign for Liverpool. Wherever you go everyone knows Liverpool FC and I am just very happy to be here," Wilson told the club's official website.
"This is a massive challenge for me. This is one of the best leagues in the world and I will be playing for one of the biggest teams in the world.
"Coming here from Scotland, people may say I might not be ready for this, but that is a challenge I want to put on myself and that's the reason I came here.
"Everyone knows about the history of this football club with 18 league titles. It wasn't a difficult decision to come here as Liverpool are not a team who come in for you every day, so I can't wait to get started."
Wilson is Liverpool's third signing in the summer after former Standard Liege forward Milan Jovanovic and free agent Joe Cole.