Abortion is the purposeful termination of a pregnancy.
For those people who believe a person dies when the heart stops beating, it stands to reason that a person begins with the heart starts beating. From this belief, they argue that any abortion which occurs after a heartbeat can be measured, is killing a newly developed person, and, therefore, murder. This would be a reasonable argument if it weren't based on ignorance of human physiology.
Western culture has a fixation with the heart due to it being thought of as the seat of consciousness, emotion, and memory for so many centuries. This is where we get phrases like, "I love you with all my heart," and, "I knew it by heart." Of course, the medical community has demonstrated that the brain is the seat of consciousness, and the heart is merely a muscle which pumps blood. A person's heart can be stopped for a prolonged time, it can even be removed entirely and replaced with an artificial proxy, and that person can continue to live.
Doctors define "death" (at least for humans) as no longer having brain activity, so, following the logic above, a new person doesn't begin until a new brain has been formed, not a heart.
However, from my experience, the heartbeat argument is merely a canard. When I ask the people who make this claim if they would be okay with abortion before then, they almost always claim that it's still immoral because they actually believe life begins at conception, not after the formation of a heart.
Which raises the question, "then why are you using the heartbeat as the cutoff point in your argument?" I think the reason people do this is because they're employing a dishonest strategy. They know if they come out and say they want to ban all forms of abortion far more people will disagree with them, so they cater to our society's physiological ignorance regarding the heartbeat. It's a way to get their foot in the door, and most people are caught offguard by it. Of course, like anyone who's trying to get their foot in the door, it's merely the start of something much bigger