Why do Shia pray with their hands at their sides?
If you pray beside a Shia Muslim, you will see them stand with their arms resting at their sides instead of folded over the chest or stomach. This is how the Ja'fari school, following the Imams of the Prophet's family (Ahl al-Bayt), teaches the standing part of the prayer.
What Shia practice is
After the opening takbir, the hands hang down and rest on the front of the thighs, fingers together, while you recite. Nothing else about the prayer changes. This is called irsal, letting the hands down.
It comes from the Imams' own teaching. In one well-known narration, Imam al-Sadiq asked his student Hammad ibn 'Isa to pray, then stood facing the Qibla and showed him the whole prayer himself, from start to finish. He stood with his hands let down on his thighs and his fingers closed together.
Shia marja', Ayatollah al-Sistani among them, teach their followers to keep the hands at their sides in this way (Islamic Laws, ruling 1112).
Not only Shia
Many Muslims of other schools pray this way too. Many Malikis, especially in North and West Africa, keep their hands at their sides in the obligatory prayers, as do the Ibadis of Oman. Most other Sunni Muslims fold their hands. Muslims have prayed in both ways for more than a thousand years, and people of different schools pray side by side in the same mosques every day.
What to do
Pray the way your marja' teaches, and respect the way others pray. In a mosque of another school you can stand as you normally do.
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Sources: Office of Ayatollah al-Sistani, Islamic Laws, ruling 1112. Shaykh al-Saduq, Man La Yahduruhu al-Faqih, vol. 1, the description of the prayer, hadith 1 (Hammad ibn 'Isa). Follow your own marja' where his rulings differ.