The best I can piece together, as a layperson reading state statutes, is because the NH Division of Historical Records[1] was assigned a duty and a budget to preserve and disseminate information on NH history [2]. The NH Historical Marker program is a joint effort with the NH Department of Transportation, and it is the responsibility of the DOT to erect and maintain the signs after approval from the Division of Historical Records [3]. The DOT has authority to erect these signs on class I-III roads which are state owned and maintained [4].
There appears to be a separate statute governing the installation of markers along class IV and V roads which are maintained by cities and towns [5].