Speaking as someone who's volunteered at such organizations, it's true although there is a minor "yeah, but..." element to it.
Put that ten bucks in your pocket and shuffle down to the grocer's and then try to feed "four for three days" with it. Almost guaranteed to be impossible, right? Paying grocery store prices up front does make it hard if not insurmountable. Food banks, however, have connections. They can buy in the kind of bulk we can't, leveraging economies of scale. Food banks also have volunteers, which cuts down the overhead as well.
They also rely on donations of close-dated products culled from stores, and on gleaners who pick unwanted fruits and vegetables and bring them to the food bank. The worst thing about going to a food bank -- aside from the hit on one's personal pride, which is silly -- is that canned and boxed foods are most common and fresh stuff is much less common or may not be available at all in winter.
I usually give money to food banks, although as a small farmer/large gardener I occasionally have a bumper crop of something so I don't yank up the plants but grow them out and take the produce in to help those in need. Along with the food, there should also be some gentle education about how to get the most out of it. Mrs. Hippie can take a 3-pound roaster chicken and have dinner for two, pick and boil the carcass to make broth, have two meals for two of chicken fried rice, make a big pot of chicken noodle soup (three meals for two) and Princess Flapdoodle the Operatic Dachshund gets a nice crispy chicken neck.
At one of the food banks I used to help, the management told me, "Give us $1000 worth of food and we'll hand it out. Give us $1000 cash and we'll turn that into $3000 worth of food."