At the time that the treaty was agreed to by the majority of the world community in the late 1990s, America was and had been recently involved in military interventions and covert operations against both nations and terror groups. President Bill Clinton, on advice from the White House Chief Counsel, decided that signing the treaty might expose an American President, as well as unknown numbers of American soldiers and operatives, to prosecution for Crimes Against Humanity under the Court's charter. While Clinton did not reject the treaty or the Court outright, he raised the initial objections. President George W. Bush later openly refused to sign the treaty or submit to the Court's authority, but by then the US was deeply involved in the Afghanistan and Iraq wars as well as the War on Terror.
So in essence, the US did not sign the treaty because it felt legally exposed to prosecution and did not want to put American leaders or soldiers in that position.