There are three major reasons for this view.
Shi'a Brotherhood
Iran and Iraq have a very strong religious connection, namely that the majorities of both countries are Shi'a. There is also a strong historic connection between the populations of Iran and Iraq. However, all of the dictatorships (from the Umayyad's onwards) in Iraq were ruled by Sunnis who repressed the Shi'a culture, prevented Shi'a from visiting their holy sites and celebrating their holidays. Since Iraq is the one of the few majority-Shi'a states, Iran wants to build a strong cultural connection. A democracy allows the Shi'a to become the majority and this cement this relationship. (This type of relationship is similar to how the United States and Canada, which are based on similar historic circumstances have a strong relationship.)
Expansion of Iranian Influence
Iran is looking to be a regional power and for many years Saddam Hussein and Iraq in general were checking Iranian influence. Iran currently supports Hamas and Hezbollah to control politics in Palestine and Lebanon respectively and directly supports the Syrian Dictatorship of Bashar Al-Assad. If Iran supports democratic parties in Iraq that eventually clinch the seats in Parliament, they begin to have a hand in influencing the direction of Iraqi Policy and have an ally instead of an adversary on their borders.
Similar Pasts, General Positivity
Iranians like anybody else, like people to be free to express their own desires and choose their own destiny. Iran sees its government as the response to a people being repressed by the Shah and desiring an alternate route. Similarly, Iraq was held down by Saddam Hussein and the new government allows Iraqis to voice their own opinions, choosing their own destiny.
Russia sold weapons to Iran and Iraq.
Iran
Saddam Hussein was the President of Iraq who invaded Iran during the Iran-Iraq War.
Large parts of Iraq, Syria, Turkey and Iran.
Iran and Iraq.
Iran > Iraq > Israel.
Iran.
Iran never looked after Iraq.
Former name of Iran: PersiaFormer name of Iraq: Mesopotamia
Iraq
Iraq and Iran were parts of Persia
Iran. The Iran-Iraq War, waged between Iran and Iraq began in 1980 and only ended in 1988.
Yes, Iraq was at war with Iran from 1980 to 1988. This was called the Iran-Iraq war.
Iraq is further west than Iran
Russia sold weapons to Iran and Iraq.
there's no allies of Iraq because even Iran and all the Arab country doesnt want the situation to be improved.
The country of Iran is larger.