You may see some person in your community in need and your empathy prompts you to contribute to a fundraiser or whatever. Singer points out that while that empathetic instinct is natural, an intellectual approach (or what Bloom is calling compassionate) results in a better overall outcome, often wildly so.
Say you read in the local paper there is a fundraiser for $50K to buy a custom vehicle for someone recently paralyzed from the waist down. Sounds great, I want to contribute! But that same $50K could literally save 20 lives in a developing country, or it could prevent thousands of cases of blindness in children. Most of us (me included) don't have nearly as strong an empathetic response to people suffering 10,000 miles away as we do for someone nearby, someone more like "us." The suffering of people living 10,000 miles way (and will likely never meet) is just as real as the suffering as someone living five miles away. Allocating limited resources based on our emotional instincts lead to more net suffering.
That is the core idea of Singer's book. Much of the rest of the book goes about step by step debunking the usual objections which get raised ("I already pay a lot in taxes for those 3rd world people", and "Why donate to a charity in Africa? It will just end up in the pocket of a corrupt politician", and "There have always been poor people and there always will be", etc)