The access problem started just after I removed all users and tried to add them back in again. After adding them back I had given edit permissions to http:\{MYSITE}* and http:\{MYSITE}Documents/Forms/AllItems.aspx, but nobody was able access the page or library and it showed the "Let us know why you need access to this list" message.
But when I changed the user policy from central administration, by adding one user for a test, he was able to view all the folders and files beyond his limits. Can you help me to give site and folder access in the right way?
Note: When I give permissions to users for a list or a list item an additional permission 'Limited access' is automatically assigned to them. Is that causing this issue?
Much appreciated if you could help me!!