The tour is running until 17 September and will feature Payne and other members of the TeamGB swimming, diving, synchro and water polo squads, touring the country to teach youngsters the various aquatic disciplines.
Member of the TeamGB Paralympics squad will join the tour, which is visiting London, Liverpool, Glasgow, Swansea and Coventry, once the Paralympics are over.
Payne spoke to Marketing about the Olympic legacy, at the Gurnell Leisure Centre in Ealing.
She said: "Hopefully, all the kids in school are more inspired to do sports and not sit around watching TV any more.
"They've seen the likes of Jess Ennis and Rebecca Adlington win medals and they see it is possible to do that. We are not superhuman or freaks of nature – we are just normal human beings who've just trained and worked really hard for this."
British Gas is backing the tour as sponsor of British Swimming, which it began supporting in 2009 and will continue to support until 2015.
The sponsorship deal involves British Gas funding the British Swimming Intensive Training Centre in Stockport and running the 'Free Swims For Britain' campaign.
Payne said: "For us, British Gas have been amazing for the last three or four years they have been with us.
"Without their help we wouldn't be here and without their help we wouldn't have all these kids swimming."
British Gas claims its free swims campaign has resulted in 130,000 people taking free swims since April, and this number is due to rise with the campaign scheduled to run until 30 August.
The Swimming Heroes tour comes after a disappointing performance from the British Swimming squad at London 2012.
The swimmers disappointed many pundits by winning just three medals, meaning the team could face a funding cut ahead of Rio, as TeamGB focuses on sports where Britain has a more realistic chance of taking home medals.
Payne has cast doubt on the idea that Olympic athletes could be in line for multimillion-pound sponsorship deals following TeamGB's hugely successful Olympics.
She said it was "probably not" likely athletes would be in line for such large deals, but added the disclaimer that "the only people that know are the athletes themselves and it is quite a personal thing".Follow @mattchapmanuk