If something's believable, it's not too outlandish to believe — it makes sense that it's true. A believable explanation for why you're late to work might be a flat tire or a broken alarm clock.
Believable stories are convincing, and believable excuses are credible. It's believable that you once spotted a movie star at a restaurant in New York, but less believable that you then went to a carnival together and rode on the Ferris Wheel. The word believe is at the heart of believable, from the Old English belyfan, "to believe." The more common spelling until the 17th century was beleeve.