She insisted she is not the new Margaret Thatcher, describing her as "absolutely unique".
May, the longest-serving Home Secretary in a century, topped the second-round poll of MPs with the support of 199 out of 330 Tory MPs.
The decision on who will lead the country out of the EU now rests with the party's 150,000 members.
May admits "difficult times" lie ahead, but she is optimistic about the future.
She urges the party to drop the terms Brexiteers and Remainers, echoing ex-foreign secretary William Hague saying "we are all leavers now".
"We have a job to do in making the best deal we can in coming out of the EU and I am very clear that I will deliver Brexit," she said.
She also embraced Ken Clarke's description of her last week as a "bloody difficult woman".
"We [women] just get stuck in. Politics isn't a game. The decisions we make affect people's lives. I also think that right now people are looking for an honesty in politics.
"We could go through some tough times and we need to be honest with people about that."
She admitted she likes to keep her "personal life personal" but says she and her husband Philip dealt with the fact they couldn't have children and moved on.
May challenged Leadsom to sign a five-point pledge card and called for a clean campaign.
It includes a pledge not to "co-operate in any way with other political parties", including donors or members.
Nigel Farage, the Ukip leader, and Arron Banks, the party's donor, have said they support Leadsom as next Tory leader.
Leadsom has refused to rule out Farage forming part of her Brexit negotiating team.
It came as President Barack Obama dismissed claims the special relationship is being put at risk by Brexit as "hyperbole". He said Brexit negotiations should not be protracted.