**Tl;DR**: The default state of food for People of the Book is halal. But for meat, we do have more specific requirements, so we check them. Allah says in the Qur'an that mentioning His Name is a requirement, and from Sunnah we learn that the time of this is always during slaughter.

Animals killed by non-Muslims do not have the name "Allah" pronounced even if all other rules of Islamic butchery are satisfied. As a result of this, if a Muslim, knowing that it's slaughtered by a non-Muslim, eats meat, he'll be considered as having stealed the right to life of an animal (similar to stealing rights of other people) in the hereafter and this'll cause him increased torment of the grave. See this for my full answer: https://islam.stackexchange.com/a/76023