Sochiro Honda, the founder of Honda Motor Company, is credited with that quote, but he's not the only hugely successful person to believe it and espouse it. Charles Kettering, who founded Delco said, "99% of success is built on failure." Writer Elbert Hubbard said, "There is no failure, except in no longer trying."
I think Mr. Hubbard summed it up best. The word failure tends to imply finality, rather than a beginning. As adults, we don't look at it like a small bump in the road, we look at it like an insurmountable obstacle.
We should take our cue from babies. They fail constantly. Look at a baby learning to walk. They fall down every few steps. When they're not falling, they cling to steady objects to move. None of this is walking as we think of it, but none of it is failure the way we think of it either. The babies don't give up on walking the first time they fail at it. They figure out how to get up from their butt and they try again. They keep on trying over and over again until they finally learn to walk.
The only failure that can't be overcome is the failure to try. If you keep trying, you're still in the game. You're still learning the skills you need to succeed. It's at that point, that failures become mere speed bumps.