The former Republic of Ireland Under-21 international was the Robins' top scorer last season, netting 18 goals to help the League One side clinch a play-off spot.
Collins, 22, has signed a three-year contract with Hibs and says it was an easy decision to move north of the border.
"I'm really excited about joining Hibs and playing in the Scottish Premier League," said Collins.
"It will be a new challenge for me and I'm aiming to adapt quickly and to score goals for the team.
"Once the bid to Swindon was accepted, the manager showed me the stadium and training centre and it was a real eye-opener; I didn't realise just how big the club is.
"So it was a really easy decision for to make and now I'm just targeting a good run of games in the side and to score some goals."
Coventry-born Collins began his career with Aston Villa but failed to make a single appearance for the club.
He had loan spells with Darlington and Burton Albion - and a trial with Aberdeen - before joining Shrewsbury Town on a permanent deal in January 2011.
Collins scored 24 goals in 73 appearances during his 18 months at New Meadow before he moved to Swindon on a free transfer in 2012.
Collins hopes he can score goals on a regular basis for Hibs and earn a call-up to the Ireland senior squad.
He said: "That would be a dream come true if I was called up, but first and foremost I need to start well with Hibs, stay fit and score goals.
"I'm going to give everything I can to help the team to do well in the Scottish Premier League and I'm looking forward to playing in front of the Easter Road crowd."
Manager Pat Fenlon said Collins has been a long-term transfer target.
The Dubliner said: "We've been watching James since he was at Shrewsbury, so we're really pleased to be able to add a quality striker like him to the squad.
"He is a bright, lively striker and he has been brought to the club to score goals. He will add further quality to our forward line."