An honesty box is a box left outside with fruit/veg/eggs kindling (anything) in, there will usually be a sign with the 'suggested' price for the items - you take what you want & leave the correct money, or sometimes more money if you think the produce is worth it.
*In some areas such as very small villages, people will leave other things in place ie cake or biscuits.....or sometimes an I.O.U.
That is one of the thingsI miss round here, back in Cambridgeshire and all round the Fens you get all sorts of fruit, vegetables, eggs, honey, plants ... put out for sale like this. We sometimes sold honey, eggs and plants if we had a surplus as well. Cheap, fresh and good quality as well.
Obviously there is a better class of criminal there. They'd break into a house and strip it, but leave the produce and honesty box intact. Beneath their dignity and a breach of trust.