Supporters usually offer one or more of three arguments: The Second Amendment to the Constitution, the need for a last ditch defense against a tyrannical government, or deterring and preventing crime. The first of these is relevant to a U.S. court, although current courts do not seem to find it very convincing, but less relevant to the rest of us. If the right to bear arms is a good thing, we should have it even if it is not in the Constitution. If it is in the Constitution but a bad thing, then we should repeal the 2nd Amendment and, until we do, interpret it as narrowly as possible.
Two hundred years ago, the second argument was persuasive. As Cromwell had demonstrated a century and a half earlier, a large professional army could create a military dictatorship. A large militia meant that you could manage with only a small professional army. If the army got uppity, its superior quality would be outweighed by the militia’s vastly superior numbers.
In the centuries since, both the relative size of the military and the gap between military and civilian weapons have greatly increased, making that solution considerably less workable. In my view, at least, conflicts between our government and its citizens in the immediate future will depend more on control of information than control of weapons, making unregulated encryption, as I have argued elsewhere, the modern equivalent of the 2nd Amendment.
I made the theoretical argument for possession of concealed weapons as a deterrent to crime in a chapter of my (webbed) Price Theory. The empirical argument was made later in an article by John Lott and David Mustard, setting off an extended scholarly (and sometimes unscholarly) controversy. While I still find the theoretical argument convincing, the empirical question appears to be still open.
There is another, and perhaps better, argument for private possession of firearms. If the population is disarmed, protection against crimes is provided mainly by the police. People very much want not to be victims of crimes, so if protection depends on the police there will be public support for expanding the powers of the police in order to better protect us. The result is a more powerful government, which I think a bad thing. Given the current government, I would expect that argument to appeal to many people who find the first three unconvincing.