Learn Swedish: De delade med sig av maten till grannarna. - They shared the food with the neighbours.

The verb dela on its own usually means to divide or split something into parts. When you want to express sharing in the sense of letting others partake in something you have, you use the fixed reflexive expression dela med sig.

Why do we need the preposition av before maten?

The full expression is dela med sig av something. You need av to connect the sharing action to the specific thing being shared, which here is maten.

The English translation says with the neighbours, so why does the Swedish sentence use till instead of med?

Because the word med is already part of the phrasal verb delade med sig, using it again for the recipients would be repetitive. Swedish frames this action as giving a share to the neighbors, so it uses till grannarna.

Isn't sig singular? Why is it used here with the plural subject De?

The reflexive pronoun sig applies to the third person in both the singular (he, she, it) and the plural (they). Since the subject is De, the matching reflexive pronoun is still sig.