Wes Davis, often mentioned on this site, used the expression on top of this post in an email the other day and parenthetically added, “(as Paul Hollywood says).” I’m not proud to say I had to Google to find out that Hollywood is a judge on “The Great British Bake Off.” But I didn’t really have to consult any sources to realize “If I’m honest” is a characteristically British expression. It just sounds like one.
Google Books Ngram Viewer confirmed the impression. It shows the expression coming on the scene in about 1990 and always being much more popular in Britain than in the U.S.:

Thoughtfully doing my work for me, someone on the Quora site asked, “Is the phrase ‘if I’m honest’ used outside the U.K.?” Three people responded, most pithily Andrew Humphrey, who said, “Wherever it is used, it is a pointless affectation. People in the UK are very fond of such redundant and pretentious words and phrases. They use these phrases to give their hackneyed or cliched pronouncements some fake importance or profundity.”
But more helpful was Luke Proctor, who dug up examples of two American using it, thus securing NOOB status:
If I’m honest I don’t believe the world would miss me if I never acted again.
Jamie Lee Curtis, actress
Because if I’m honest, people in the white world might be appalled, but in the black world they’re making myths out of me. And I know that ain’t the life
John Singleton, director
I also found, amazingly, no fewer than eight popular songs called “If I’m Honest”: by Blake Shelton, Missy Higgins, Brendan Murray, Julia Gargano, Jay Denton, the group All That Remains, and Kaitlyn Bristow of “The Bachelorette.” I know Shelton is American and assume Bristow is; I’ll leave it to you lot to sort out the nationality of the rest.
In a post on her blog, Separated by a Common Language, linguist Lynne Murphy did some investigating and found out that not only “If I’m honest,” but also the similar expressions “If I’m being honest” and “To be honest,” are used far more in the U.K. than the U.S. She goes on to muse:
One has to wonder: why are these such popular idioms in BrE? And then one has to wonder: is it because most of the time people are expected NOT to be honest, so it has to be marked up where people are being honest? There may be something to that — the British, after all, have an international reputation for not saying what they mean.
Of the three expressions, the one that sounds most familiar to my American ears is “To be honest.” So I plugged it in to Ngram Viewer and found this:

That is to say, it was roughly equally popular in both countries for a long time, and was used markedly more frequently in both between about 1980 and 2000. After that, it skyrocketed in Britain.
Why? If I’m honest, I have no idea.