One of the conditions for meat to be halal is that the one slaughtering the meat is a Muslim. The verse you mentioned is referring to the non-meat items such as vegetables, fruits and grains as was explained by our 6th Imam (as):

Hussain bin al-Mundhir said to Imam Ja’far as-Sadiq (as): We are a people who frequently go to the mountains and the distance is great between us and the mountains. We therefore buy animals in large number for food, and we ask the herdsmen about their religion and they reply that they are Christians. “So, what do you say about the slaughtering of animals by the Jews and the Christians?” The Imam said, “O Hussain! The Islamic slaughtering can be done with Allah’s name only and no one can be trusted with that except the people of tawhid (i.e., Muslims).”