Well I suggest that you will create a new Winforms application. Then drag Checklistbox and button into form and arrange them.
Then double click form at any area except over buttons and checklistbox.
Now you should have a function which is called when application is loaded (this way you dont need another button to load files into checklistbox).
Now use Directory.GetFiles method to get all files form certain directory.
Checklist items are added with method CheckedListBox.ObjectCollection.Add(...)
Now go back to designer mode where you see a form, checklistbox and button. Double click button to create a event handling for button click.
Then use foreach loop to iterate through all checked items from checklistbox objectcollection and call File.Move for each item which is checked.
Use full absolute file paths, they are easy to use in this scenario.
File Move: https://msdn.microsoft.com/en-us/library/system.io.file.move(v=vs.110).aspx
Get Files: https://msdn.microsoft.com/en-us/library/07wt70x2(v=vs.110).aspx
Similiar question and some answers:
Find all files in a folder