In depth article, but here is the conclusion if you don't want to read it all.
At least read the opening at the link below.


"Bleeding heart" was revived in a political context in 1954, by another infamous right-winger, Joe McCarthy, who called Edward R. Murrow one of the "extreme Left Wing bleeding-heart elements of television and radio." It wasn't until the 1960s that it really started to come into common use, though. In 1963, the satirical columnist Russell Baker put it on a list of political insults: "If one is called a 'phoney,' about the only thing he can do is come back with some epithet like, 'anti-intellectual' or 'bleeding-heart liberal'...or 'you must be one of those peace nuts.'" By the end of the decade, Ronald Reagan, then newly elected governor of California, had picked it up as a way to describe his political trajectory. "I was quite the bleeding-heart liberal once," he told Newsweek. By 1970, he was known as a "former 'bleeding heart' Democrat."

After that, the phrase was fully ensconced in political short-hand and quickly claimed by liberals as a positive trait.  "You are called a bleeding heart liberal because you have a heart for the poor," one told the Times. "Count me with the bleeding heart liberals," an NAACP lawyer wrote in a letter to the editor.

https://www.atlasobscura.com/articles/origin-bleeding-heart-liberal

Is it because they don't use their brain?