Eating locally-produced food helps the climate problem by reducing the use of fuel to transport raw materials to produce the food. When food is produced locally, there is no need to transport food it to be consumed by the community. This will definitely decrease the burning of fossil fuels and greenhouse emissions. There will also be a decrease in using packaging materials if you do not need to haul food for long distances because it is available locally, and so there will be lesser waste materials to dispose of decreasing pollution.
Buying locally grown products can improve the local economy by providing jobs. This will also build better human connections within the community, and promote social responsibility as we feel more connected to each other and become responsible keepers of our community. 
This will also lead to better city planning and the utilization of space through converting the unused land into community gardens while promoting a healthy living environment in the community and promote more "green space".
In my school, we have a Living Lab which is a ve getable and botanical garden and a certified wildlife habitat which also provides fruits and vegetables the students consume during lunch time. The Living Lab teaches students about sustainability and the conservation of resources, plants, and our environment as a whole. While carbon emissions are rampant in the industrialized cities of the Bay Area, green areas such as the Living Lab, have trees and vegetation which utilizes the carbon from the atmosphere and converts it to oxygen while supplying our school community locally grown produce.

I absolutely agree with you, Gabriel. Both in the Bay Area and around the world, there are programs called CSAs. CSA stands for community-supported agriculture, which is a method of connecting the community with the food they eat. Programs like Full Belly Farm and Imperfect Produce, to name a few in the Greater Bay Area, not only give people access to organic fruits and vegetables but also make it their mission to show the communities they serve exactly how the food gets from the farm to their table. A lot of these programs have events where people are invited out to the farms themselves so that they can experience what it's like to work in agriculture and develop a deeper connection with their food.

Source: http://www.justfood.org/csa