The 29-year-old turned down a new deal at beaten League Two play-off finalists Exeter to join the third-tier Pilgrims.
"Not everyone will understand my reasoning but those who do, hopefully they understand," he told BBC Devon.
"The club offered me a deal which we tried to negotiate and it just didn't end probably the way it could've."
Grant joined Exeter on a two-year contract in August 2015 after the club's fans raised £30,000 towards his wages.
He has not played for Jamaica since May 2016 when he scored in a friendly victory over Chile.
"I wouldn't tell any lies, I thoroughly enjoyed my time [at Exeter] and was very close to staying, but it's football," Grant added.
"I made the decision to join Plymouth and now I look forward to the challenge.
"It's [the move] definitely something I had to think about, but I'm delighted to be here now and hopefully I can have a good start."
Jamaica international forward Joel Grant says he came "very close" to staying with Exeter City before joining their Devon rivals Plymouth Argyle.