It's true that you often pay more for locally grown things. The reasons for that are fairly simple really - small farms are not subsidised by government the way large ones are, they typically have to pay for all their costs unlike huge farms that off-load them to society at large, vertical integration, keeping animals in immoral conditions and devestating the soil to save costs, etc.
It was not long ago that food costs were a significant portion of our budget - it took about 14 families to support one farmer - now it is 10 times that amount. People are going to have to get used to the idea that food, a basic necessity of life, is worth spending money on, and farmers deserve to get paid for their work.
There are a number of things to do to save money on food while also eating locally and sustainably. Eating seasonally as Rus suggested is a big one. In some areas it is difficult because locally produced veg are not easily available at stores or markets in the winter. One possibility is to find a CSA arrangement that runs in the winter. If that isn't available, the best option is to buy a freezer and in the produce season, freeze and preserve things for winter. It is also not difficult to build a small cold room for potatoes and similar crops. Properly frozen food is typically more nutrient rich than imported fresh food, btw.
As far as meat - the main solution is to eat less, and different types. You want to be buying "grass-fed" beef, and if possible pasture-raised pork. Cheaper cuts are easy to learn to cook, as well as less used parts like liver, kidney, oxtail, trotters, etc. If you buy a meat box in the Fall, you can often get it at reduced cost if you give up some of the popular cuts like chops oven roasts and ask for them to be replaced with things like hocks or pot-roasts.
Eating less in a meal can mean instead of using a chop or cut of meat for each plate, make things that incorporate only a small amount of meat - add a bit of chopped ham to a stew with some of the peaches you canned in the summer, for example. For most of human history meat has been eaten in significantly smaller quantities than today.
Old cookbooks can have great ideas for these kinds of meals, especially old war cookbooks. (One of my favorite war recipes is something called a "Buckinghamshire clanger" which uses biscuit dough, potatoes, onions, and a tiny amount of bacon.)
And growing a garden is absolutely the best solution. A suburban yard can actually grow a significant amount of your requirements, and even in cities where there is no yard community gardens are often available. The main thing is to start small so you don't get overwhelmed.
