Question

what is one reason European governments protect the growing of food with subidies?

Answer 1

Unlike tariffs and quotas, no consumer surplus is lost, because price doesn't increase. The subsidy shifts the domestic (European) supply curve to the right, and leaves everything else unaffected.
If your question is asking for one reason why they protect the agriculture industry (not necessarily with a subsidy), it is because the European governments want food security i.e. if hostilities arise and food imports are cut, they don't want to starve.