How is Iran able to project its influence so far away from home, even though they are under US sanctions?
First, most of the international sanctions, not just US, have been lifted in the wake of the JCPOA, usually called the Iran Nuclear Deal.
Iran is the center of Shia Islam, as opposed to Sunni Islam. Shia and Sunni correspond roughly to Persian and Arab, although the Muslims who aren't in either ethnic group tend to be Sunni, and there are a few cases of Persians who are Sunni and Arabs who are Shia. Lots of longstanding grudges between those two groups, dating back to the basic argument over who should lead Islam after Mohammed's death. Shia believe it should be a direct descendent of Mohammed, while Sunni believe it should be an elected position. The Shia have a bit of a problem with the Sunni, as Sunnis control the holy city of Mecca in Saudi Arabia.
Before that, we have Arab vs Persian, quite possibly the most longstanding grudge in human history. They really don't like each other. Never have, and probably never will.
Iran is active in Syria, Yemen, Lebanon, and Iraq, as those countries have large Shia populations, or in the case of Yemen and Syria, Shia aligned (Houthi and Alawite).
Iran is more closely aligned with Venezuela, which has a decaying economy and few friends, despite Venezuela's huge oil reserves, while both nations have a bit of a pariah status with the economic powerhouses.
Iran is simply pursuing a course of action that it sees as in its best interests. It is not really projecting power, other than very limited military support to Syria and Yemen, and while the Sunni-Shia conflict raged in Iraq, to the Shia majority there.
One other factor to be considered, is our current 'news media' and their need to stir up panic to raise ad revenue. They tend to make a mountain of a molehill on a lot of stories... good for their business, until the general population figures that out.
As mentioned in another answer, Iran isn't necessarily "projecting power."
It might be more logical to say it is simply seeking allies that can help it stand up to the U.S./Israeli empire. Other countries similarly seeking to free themselves from "Yankee imperialism" would therefore make logical allies. In recent years, there have been loose alliances between Cuba, Venezuela, Argentina, Libya, Iran and Russia.
The alliance has weakened with the destruction of Libya and the death of Venezuela's Hugo Chavez. The U.S. has similarly made great gains in destabilizing Latin America, particularly Venezuela and Argentina (which recently elected a right-wing president). Syria has been largely neutralized as well, and it isn't clear whose side Russia is one.
So Iran isn't "projecting power" the way it once did.
Nevertheless, it is still a very strong regional player and remains a major oil producer. There are millions of people around the world who resent the United States' power and arrogance, and such people automatically tend to support countries like Iran.
Those operations are reasonably cheap for an economically advanced state like Iran.
Iran prioritizes its regional presence and might make budget cuts elsewhere to continue to further its international goals.
Little of its defense industry has been linked to the west in recent decades, so sanctions from the US are not as meaningful to power projection as they are to quality of life of the people (McDonald's aren't called McDonald's).