IPA: /ˈɡrænts.mən.ʃɪp/
KK: /ˈgræntsˌmənʃɪp/
The skill or ability to successfully secure funding or grants from organizations or government bodies.
Her grantsmanship helped the nonprofit secure several important funding sources this year.
Grantsmanship is formed from "grant" (from Old French *granter*, meaning to give or allow) and "-smanship" (a suffix denoting skill or art). The term refers to the skill or art of obtaining grants or funding.
Think of the skill involved in 'granting' something — that's why grantsmanship means the art of obtaining grants.