To plow new fields, you must have enough clear space on your farm for the new field and you must have coins. Each plowed field will cost you 20 coins and earn you 1 XP (experience points are what you need to increase your game level and earn some free Farm Cash), When you click on the Plow tool it picks it up. As you and move it over your farm the background of the plow field will be green if there is space for you to plow a new field. If the background is red, there is not enough space.
To stop using the Plow tool click on the green man.
Click on the Preferences icon and add the check mark to Remove Spaces Between Fields then click on the green check mark to save that change. Existing Fields which are already plowed or harvested must be deleted first. Then you will be able to plow new fields so that they are closer together without those spaces. This picture shows the difference between plowing without spaces and with spaces. As you can see if you plow without spaces you can fit more fields into roughly the same area.