The Chelsea striker, 19, has joined on a season-long loan and hopes to make a similar impact in the Premier League than he did in the Championship for Bristol City last year, scoring 26 goals.
Fernando Llorente will miss the opening game at Southampton on 12 August, which could see Abraham get an early chance.
"I can't wait, I'm so excited right now," Abraham said.
"I came here to score goals and help the team as much as I can... it's good to score goals, it lifts your confidence going into the season especially and I'm looking forward to it.
"I'm going to prove to the gaffer [Swansea manager Paul Clement] why he brought me in... and I'm going to do my best and try as hard as I can."
Abraham, who scored for England in the semi-final of the Euro Under-21 Championship in June, already has two goals in three pre-season games for Swansea.
While he starred for Bristol City last season on loan, scooping the club's Player of the Year, Young Player of the Year and Top Goalscorer awards, he realises it will be a big step up to the Premier League.
"Of course it's a different level, this is the best league in the world so I know I'm going to come across some world-class players," said Abraham, who signed a new five-year deal with Chelsea during the summer.
"I just want to do the best I can and hopefully to beat last year's goal tally, by all means that's what I'll go for.
"It's a fantastic squad and I'm playing with players I grew up watching, so it's a dream come true.
"I have to show the fans what I'm truly made of and what kind of player I am.
"I score goals but I've got good feet, good pace as well, so I would like to show the fans and help the team as much as I can while I'm here."
Tammy Abraham is aiming to prove his worth to Swansea City with goals.