It's actually deceptively simple: There are two major issues at play, but it's primarily an ideological issue between the Wahhabi (Sunni) Saudis and the Shiite Iranians, with specific events in recent history adding fuel to the fire. Think of the religious wars in Europe in the aftermath of the reformation. There are obvious differences, but it's similar enough in terms of the animosity that developed between Catholics and Protestants to help one understand. Iran came to be seen as a haven for Shiites, and a powerful ally against the, frequently Sunni, majorities. As an example of the animosity, in 1987, during the Hajj, a violent incident happened in which 400 Shia - two thirds of them Iranian - demonstrators were protesting against the United States and Israel and died under disputed circumstances. Iranians claimed the Saudis fired on the protestors, and the Saudis claimed it was the result of a stampede (the bodies were displayed to Western reporters with bullet wounds after they returned to Tehran). The Ayatollah Khomeini, still fighting the Iran-Iraq war, called on Muslims to overthrow the Saudi regime, and the Saudis responded by withdrawing their ambassadors and severely cutting the number of travel visas granted to Iranians. Iran in turn boycotted the Hajj until 1991, when both restrictions were eased in the wake of the Iraqi invasion of Kuwait.

Secondarily, because of it's unpleasant history with the Qajars and Pahlavis, Iran has been strictly anti-Monarchy since the Revolution with royalists frequently imprisoned. The royalist political party - the Constitutionalist Party of Iran - was founded in Los Angeles in 1994 for this reason. The formation of an Islamic republic threatened the legitimacy of the Saudis, who are absolute monarchs. They feared it would lead to a rise in republican sentiment throughout the Arab world. When the Iranians began funding Shia militant groups aiming to overthrow Sunni governments, it was seen as justification of that fear. Also after the Iranian revolution, because of the US-Saudi alliance, Saudi Arabia was characterized by Iran as an American pawn aiming to destabilize the region for their own gain, or a servile state dependent on Western power for their existence. There's also the Saudi funding of Hussein in the devastating Iran-Iraq war, which cost a half-million Iranian lives, destabilized the country politically, and absolutely destroyed the Iranian economy. The Iran-Iraq war is an extremely important event in the modern history of Iran, especially politically. It illustrated how isolated Iran was, and ended as a humiliation to Khomeini.

There are more examples of the low relations (and some attempted improvements), but the causes boil down largely to issues concerning religion, regional influence, and governmental incompatibilities, with the magnified effects of tragedies and diplomatic insults making things worse.