Several factors:

1. Maliki (Shia Arab leader), Talabani (Kurdish Leader), Barzani (Kurdish Leader) and many other of todays Iraqi leaders all spent many years, if not decades, of their lives in exile in Iran as political refugees/under protection, during the Saddam period.

2. The Shia clergy and establishment have two main centres, Qom (in Iran) and Najaf (in Iraq). So there is a strong religious link. Of course, there is a difference in views on level of interference in politics but that (velayt-e faqih) is not a Qom vs Najaf issue but more something a group of political clerics prescribe to in Iran. The point is there is a strong religious connection, centuries of connection/communication.

3. After the US dismantled the Iraqi army and things turned to chaos, Iran poured huge amounts of money, investment, aid, propaganda, military support, rallying etc.. of groups and militias. Iran understanding the local society better gave it an upper hand as the US struggled to contain things. This gained them huge long-term influence. So much so that today, the US can't convince Maliki or many other influential iraqi figures to do anything, yet Iran can.