It depends. But here are some basic steps.
First, the soil is turned or loosened. A farmer uses an implement that is hooked up to his tractor to do this. But a someone with a small garden could use garden tools. Sometimes compost is added to help the soil stay loose and moist. Compost also contains the minerals that the carrot will need to grow and stay healthy.
When the seed is planted in the ground, it can take about 3 weeks for germination to occur. (Germination is when the outside layer of the seed breaks open and the very beginnings of the carrot happen.) If there is not enough rain, the farmer will have to water the ground where the seeds are planted to keep them moist.
When the carrots are ready to be picked, a big machine is used to loosen the roots, (or a person can loosen the roots with a big garden fork). Then workers gather the carrots from the field and bring them back to the packing shed. They are cleaned up and sorted, then sent on their way to your CSA box.