It seems that he just became more conservative over the course of his life. Perhaps this is best answered in his own words:

Asked why he, an ardent New Deal liberal in his youth, had turned away from the Democratic Party, Reagan said simply: "I didn’t leave the Democratic Party. It left me."

A funny anecdote on the subject:

On a rare slow news day during his Presidency an NBC news crew, following a White House meeting of GOP and Democratic leaders, walked around the great now-empty table and began noting the doodles that each leader had made on a notepad. Democrats, the reporter showed, mostly had drawn doodles of animals or human faces. Republicans had drawn geometric forms, squares within squares or the like – with one exception. Republican President Ronald Reagan had drawn faces and animals. He doodled like a Democrat!

The answer can be found in Reagan's 1964 convention speech, "A Time for Choosing":

"As a former Democrat, I can tell you Norman Thomas isn't the only man who has drawn this parallel to socialism with the present administration. Back in 1936, Mr. Democrat himself, Al Smith, the great American, came before the American people and charged that the leadership of his party was taking the part of Jefferson, Jackson, and Cleveland down the road under the banners of Marx, Lenin, and Stalin. And he walked away from his party, and he never returned to the day he died, because to this day, the leadership of that party has been taking that party, that honorable party, down the road in the image of the labor socialist party of England."