This phrase refers to the biological response prompted by excitement: increased pulse rate and breathing, particularly among those who innately admire Richard Cory.
The poem tells us that Cory was incredibly rich and elegant, and that he was a man of great refinement. The combination of all those factors was what caused pulses to "flutter" when people saw him about.
The poem also shows us the great price of wealth, however: At the end, despite Richard Cory's lavish lifestyle, he winds up taking his own life. This makes the reader ask the question -- were people's perceptions of his greatness accurate? Perhaps people's pulses should not have "fluttered," after all.
When a person feels excitement, their heart beats faster, quickly and lightly - it "flutters". When the author of the poem says about Richard Cory that "he fluttered pulses when he said 'Good morning'", he is indicating that Mr. Cory was such a celebrity that people would get all excited if he greeted them personally. Think about your favorite rock star, or anyone really famous that you would like to meet. Imagine if you passed this person on the street one day, and he or she should stop, look at you, and say "Good morning". Think of the excitement you would feel if this should happen!
People looked at Richard Cory that way. He was someone about whom they would read in newspapers, someone glamorous, and very rich. The sad thing was, "he was always human when he talked", but no one could see that through all the "glitter" and the hype. Even though people would have loved to meet him, and, if he recognized them, their hearts would be "all a-flutter" because he was so famous, no one really actually knew him as a person, and he died a very lonely man.