The young striker moved to the Cherries in an eyebrow raising £17.5 million transfer in January.
He had not scored a Premier League goal for Liverpool before the move.
But Boothroyd believes Solanke has the talent to be successful at the mid table Premier League side.
The England under-21 manager told the Daily Echo: "I am glad he got his goal and he does clearly love playing for his country as well as his club.
"It was nice for him to wear the armband because he is a leader – he's a quiet leader.
"He is a good guy and is one of those people who doesn't use 20,000 words when one will do.
"He doesn't say an awful lot but, when he does, people tend to take notice of him.
"He has some competition with Callum (Wilson) and Josh (King) but I was really pleased when he got transferred here (to Cherries) because I thought it was really good for him.
"He will get the opportunity to show what he's all about with a good club and a good coach.
"Hopefully when he gets to us, he's got a bit more game time and will be able to hit the ground running."