Answer by taking a quote from the following article:

Palmer was born to Doris (Morrison) and Milfred Jerome "Deacon" Palmer (1905-1976) in Latrobe, Pennsylvania, a working-class steel mill town. He learned golf from his father, who had suffered from polio at a young age and was head professional and greenskeeper at Latrobe Country Club, which allowed young Arnold to accompany his father as he maintained the course. Palmer attended Wake Forest College on a golf scholarship. He left upon the death of close friend Bud Worsham (1929-1950) and enlisted in the U.S. Coast Guard, where he served for three years, 1951-1954.

Palmer's early "fear of flying" was what led him to pursue his pilot certificate. After almost 55 years, he logged nearly 20,000 hours of flight time in various aircraft. His personal website reads:  Next to marrying his wife, Winnie, and deciding on a professional career in golf, there's only one decision Arnold Palmer considers smarter. Learning how to fly an airplane.  On Palmer's 70th birthday in 1999, Westmoreland County Airport in Latrobe was renamed Arnold Palmer Regional Airport in his honor. According to their website: "[The airport] started as the Longview Flying Field in 1924. It became J.D. Hill Airport in 1928, Latrobe Airport in 1935 and Westmoreland County Airport in 1978. Complementing a rich history rooted in some of the earliest pioneers of aviation, the name was changed to Arnold Palmer Regional in 1999 to honor the Latrobe native golf legend who grew up less than a mile from the runway where he watched the world's first official airmail pickup in 1939 and later learned to fly himself." There is a statue of Palmer made by Zenos Frudakis, holding a golf club in front of the airport's entrance, unveiled in 2007.  Palmer thought he would pilot a plane for the last time on January 31, 2011, and flew from Palm Springs in California to Orlando in his Cessna Citation X. His pilot's medical certificate expired that day and he chose not to renew it. However, public FAA records show he was issued a new third-class medical in May 2011.

How did Arnold Palmer become a pilot?
Palmer's early "fear of flying" was what led him to pursue his pilot certificate.