That's an interesting question, which in the course of human history has meant those with the might have the "right". The US and other western nations have done this throughout the world, and in this particular region, their actions are what led to much of the mess it's in now.
Exert influence, much like other nations have done. Iran is a Shia nation surrounded by mostly Sunni nations, so it's been keen on empowering those who subscribe to that form of Islam, who have been suppressed minorities in Sunni Muslim nations. One can critique the means, but it is the same desire other powerful states seek to exert their influence in their region.
I don't have an accounting of who is the regional leader of sponsoring terrorism in the region, but the main players there all do. The small monied Persian Gulf states have (and do), and the Saudis most certainly have, with one of those groups (Al Qaeda) leading the largest terrorist attack on US soil.
Their position is one against the invasion of European Zionists into Israel and dispossessing the Palestinians. I say this to answer your question, and not to justify their position. This was the view of many of Israel's Arab neighbors because the founding of Israel contains this hard fact along with the actions of Arab nations to attack Israel as well. Over the years Iran's response has been to counter the growth of Israel's power, and since it cannot achieve that directly though military force and technology, it applies its pressure through the various groups it uses as proxies.