"A person who is considered excessively sympathetic toward those who claim to be underprivileged or exploited."
"Of, relating to, or characteristic of liberalism."
" A political theory founded on the natural goodness of humans and the autonomy of the individual and favoring civil and political liberties, government by law with the consent of the governed, and protection from arbitrary authority."
So from this, we can gather that a "bleeding heart liberal" is an individual that aligns themselves with the Liberal party(aka left), and is particularly sympathetic to the underprivileged class. Oftentimes this phrase is misused to denote any far-left individual. Ironically, some far-leftist are anything but liberals at all, but most have some bleeding heart tenancies.
"That bleeding heart liberal jury let a rapist go with six years in an institution."
"The Governor got re-elected by pushing his bleeding heart liberal views, and got the sway vote from the lower-class."