I went off beefburgers after I got food poisoning from a take-away.

Please explain the highlighed words.
In your sentence it is referring to the shop that sells this food. They are commonly called 'take-aways'.

Again, I suggest using a dictionary.

"takeaway UK noun [ C ] (US takeout)
a meal cooked and bought at a shop or restaurant but taken somewhere else, often home, to be eaten, or the shop or restaurant itself"

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In your example takeway means - prepared food bought for eating elsewhere
 nona the brit's reply was promoted to an answer.
Hi nona,
First, I wrote the same as you, but then I changed my mind and I wrote what you can see above.

Why shop, NOT food from the shop?

You get food poisoning from the restaurant / bars/ cafes/shops but you are poisoned by/with sosages or maybe by toxins from food, am I right? Is that the explanation?
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Now I've looked at it again it could be either the shop or the food. Emotion: big smile