If by "state taxes paid", you mean MN state tax withheld, from your pay (box 17 of your W-2), you file a MN state tax return to claim the refund.
A member of the Minnesota National Guard or other military reserve unit is
allowed to subtract pay received for active service. “Active service” includes:
• certain state active service, such as assistance in natural disasters and
searches for lost persons (Minn. Stat. § 190.05, subd. 5a, cl. (1));
• federally funded state active service, under U.S. Code, title 32 (National
Guard), such as, weekend drills and annual training (summer camp),
special school attendance, airport security, or active duty for special
work (ADSW) (Minn. Stat. § 190.05, subd. 5b);
• federal active service, under U.S. Code, title 10 (Reserves), such as
weekend drills, annual training (summer camp), special school
attendance, pre- or postdeployment-related duty, and time on medical
hold under U.S. Code, title 10; active duty orders while recuperating
from an injury; and Active Guard and Reserve (AGR) service under
U.S. Code, title 10 (Minn. Stat. § 190.05, subd. 5c).
Beginning in tax year 2014, “active service” also includes service by Minnesota
residents working in AGR status under U.S. Code, title 32 (federally funded
state active service), and service by current or former members of the National
Guard or reserves ordered to active service by the adjutant general to perform
full-time administration of the Department of Military Affairs.