We see things through the lens of our experience. Since I have seen numerous fields only plowed around the edge, I thought that was the OP’s question. The firebreak makes sense. So did repairing the compactation from the grain trucks only driving on the edge near the road. I also have heard forming a barrier to weeds and insects.
I think he meant to ask why fields are plowed but not planted along the fence.
I think the OP is an opening line to a joke and soon the OP will return to give us the punchline.
While we wait, I have another farm joke.
Why does a chicken coop have two doors?
Because otherwise it would be a chicken saloon? (Just a guess.)
If it had four doors, it would be a chicken sedan.
Winner! 
It may allow annual weeds to grow, but it prevents brush and saplings getting established. Without intervention, fields turn to woodland.
The fields I see are in the fall and are planted the next year.
IANAF (far from it) but I believe those are chemical/pesticide breaks.
Rain/water runoff carries residual chemicals onto roadways, streams and other places that may end in the water table.
Some time ago a friend of mine told me that the EPA found elevated chemicals in the streams adjacent to farmland, and so the farmers were required to widen the area between the crop and streams.
In areas where flood/furrow irrigation is used - mostly on high-value row crops - many farmers use a “filter strip” at the lower end of a field and sometimes around the perimeter as well. Generally, this is just a +/- 20 ft. swath of come cheap cover crop, often wheat or annual rye, laid down to trap sediments and debris from runoff water before it is carried away into the drainage system, which in turn usually flows into perennial streams. In addition to clarifying the runoff water and keeping topsoil on the field from whence it came, this also traps some ag chemical residue which may be fixed to the soil particles rather than dissolved in the water. Not sure if this is what the OP is referring to, however.
SS