In religious areas of Jerusalem you can often see double plastic-wrapped bags of leftover bread or challah next to garbage dumpsters. Rather than throwing the bread in the bin, people prefer to place it where someone poor might come and take it.
There is a strong aversion to throwing out bread, and particularly challah, in Judaism. People will make it into French toast, use the crumbs for stuffing, put it out for the birds, leave for the poor or wait for it to go stale and mouldy before throwing it away. The traditional sources for this are quite thin. There’s a hint in Job 15:23 (“The wicked man... wanders abroad for bread, saying ‘Where is it?’”) and a story in Talmud Chullin 105a which suggests poverty can result from throwing bread on the ground.
Above all, the practice of not throwing away bread expresses a strong sense that bread, and particularly the bread we eat on Shabbat, represents God’s blessing to us. We want to treasure and show our gratitude for all of it and not waste or disrespect the slightest bit.