Why do farmers plow the field before sowing the seeds?
They don't. Plowing is done only to break up pastures or hay fields that are being converted into crop land. The cultivator, not the plow, is the main tillage machinery that is used to break up harvested fields in the first step to preparing the soil for seeding. The plow is NOT the only tillage machinery that is used to till the field, unlike that which is used to "prep" fields on farm games like FarmTown or Farmville, which tend to have a lot of discrepancies in respect to how agriculture REALLY works.
After plowing is done, a disc needs to be put on next to break up the massive clumps of earth turned up by the plow. Often one or two passes are needed to completely break up the clumps. The disc can also be used after the cultivator is used on the field. Often the cultivator needs to be used again in the spring time when anhydrous ammonia is put in the field prior to seeding. After the soil is fine enough for seeding, only then is the seeder used to sow the seed to be harvested in the fall.