It is impossible to know from your photo why that part of your arborvitae is turning brown. If it is not a large area, just go ahead and prune it out.
If it is a larger area, you will need to look carefully inside the shrub at the branches and around the trunk area to see if you see any signs of injury or holes or cankers.
They resent drought, and we have been very dry.
If it's at the base, it could be dog urine damage.