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."

An Institute of Directors report hailed the business model as a blueprint for success and indicated that the UK could learn from the Manx example.
According to government figures, the estimated value of the Isle of Man space industry for the three years to 2013 will reach Â£1.1bn.
Many investors have been attracted by the 0% rate of corporation tax.
The report, published in London on 18 May, said: "The Isle of Man shows how quickly a powerful niche can be built by a very small player."
It adds: "In the new space economy, you can be small and succeed. You don't need astronauts to be in the space business."
There are currently 30 space companies located in the Isle of Man out of a international total of 54.
Jamaica international forward Joel Grant says he came "very close" to staying with Exeter City before joining their Devon rivals Plymouth Argyle.
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The Isle of Man has been praised for "building a powerful niche" within the global space industry.