A landlord is allowed to inspect anything that has to do with the home or apartment to be sure the place is in sound and safe livable condition.
The landlord also inspects for any damage done to the apartment or home.
The landlord will look for mold, water damage, damage done by the tenant, inspect the plumbing, inspect the doors, toilet, shower etc.
A landlord may also want to look into your closet to be sure the walls in the closet are in good condition and to make sure there are no water damage or mold in the ceiling or walls.
The landlord should not go through your belongings though but they may ask you to move the clothes or anything out of the closet for the inspection so that they can tell if your closet has any of the mold or other issues.
A landlord is only allowed to show up unannounced without advanced notice for emergencies and enter the rental unit to do emergency repairs.
However the landlord can show up without announcing it to you but they cannot enter without advanced notice unless it's an emergency.
That means that for inspections of the rental the landlord must give you 24 to 48 hour notice before they come by to enter the apartment or house to inspect.
The landlord should not go through your things and will just look over the rental unit to be sure no damage is done and to make sure everything is okay with the apartment or home.
So if you have anything that you do not want the landlord to see you should put them in your car or hide them in the closet.
The landlord might look in the closet but most will not and will just look in the rooms and take note of any possible damage etc.
A landlord inspection should take place every 6 months or once per year.