State highways are constructed and maintained by state governments and serve as major roads within a state.
Option A: Refers more to national highways, which facilitate inter-state connectivity.
Option B: Such connections are typically handled by national highways or expressways, not specifically by state highways.
Option C: Correct. State highways connect the capital of a state to its various district headquarters and important towns within the state.
Option D: This is again a function of national highways.
So, the correct answer is (C): To connect state capitals with district headquarters.