The conditions for halal slaughtering
Basically food is made of some ingredients among them you may find meat or flesh so the general ruling allowing to eat the food of the people of the book in (5:5) allows eating the meat of slaughtered animals (as long as they are not among those quoted in (5:3), (2:173), (6:121)) according some conditions the most important is that slaughtering is similar to our slaughtering here the conditions from this fatwa on islamqa:
- That the meat be slaughtered as a Muslim does it, but cutting the throat and oesophagus and letting the blood flow. If the animal is killed by strangling or electric shock or drowning in water, its meat is not permissible. Similarly, if a Muslim does that, the meat is not permissible.
Now to the possible confusion created by saying it is not allowed to eat from a slaughtered animal one which the name of Allah is not mentioned.... this isn't exact. The point is as you may read in (2:173) that this is meant in relation to pagan worship, so animal slaughtered by Christian or Jew is allowed as long as they don't dedicate it to some other deity or a person:
- Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said:
What is meant here is that over which a name other than that of Allaah is mentioned at the time of slaughter, such as saying “in the name of the Messiah” or “in the name of Muhammad” or “in the name of Jibreel” or “in the name of al-Laat” and so on. End quote
This is because of the sahih narration on the authority of the mother of believers 'Aisha:
Some people said, "O Allah's Messenger (ﷺ)! Meat is brought to us by some people and we are not sure whether the name of Allah has been mentioned on it or not (at the time of slaughtering the animals)." Allah's Messenger (ﷺ) said (to them), "Mention the name of Allah and eat it." (Sahih al-Bukhari)
So if they don't dedicate it at all it is fine too!
In Sunan abi Dawod you may find that 'Aisha mentioned that those people were new to Islam!
So verse (5:5) allows the food of people of the book and the food made of slaughtered animals by people of the book as quoted in the above fatwa.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said:
... The Prophet (peace and blessings of Allaah be upon him) ate meat slaughtered by the Jews and did not ask questions.
The evidence for accepting it from Muslims and ahl-al-Kitab
However what exactly says that the slaughterer must be a Muslim, Jew or Christian can be concluded from (5:3) beside the the context of (5:5):
Prohibited to you are ... except what you [are able to] slaughter [before its death], and those which are sacrificed on stone altars, ...
Here the Arabic words ذكيتم dhakaytum and ذبح dhobiha have been used to make a difference between a pagan sacrifice and a "correct from the perspective of shari'a" slaughtering it is used in the hadith in this meaning too:
"The slaughter of what is in the womb is included in the slaughter of the mother if it is perfectly formed and its hair has begun to grow." (al-Muwatta', Sunan ibn Majah, Sunan abi Dawod and Jami' at-Tirmidhi)
And can be understood this way in tafsir al-Qurtobi of this verse:
In the 7th, 8th, 9th and further items he considered as worthy to discuss on the this verse.
Here just the linguistic issues:
8th item (الثامنة)
ذكيتم الذكاة في كلام العرب الذبح
dhakytum ad-dhakata in the speech of the Arabic people means slaughtering
9th item (التاسعة)
ذكيتم الذكاة في اللغة أصلها التمام
ad-dhakat in Arabic originates in "the completeness"
Therefore in the same fatwa you may read:
Shaykh ‘Abd al-Rahmaan al-Barraak (may Allaah preserve him) said: “...
If those who produce the meat are not Jews or Christians, then the meat that they offer is haraam. ...
For more details read the fatwa I was partly quoting and my other Arabic references islamway and islamweb.
As Tamer quoted most of the verses in his answer I've not quoted nor linked them if they were available there.
See also the Arabic wikipedia article on ذكاة linked from Medi1's comment.