Question: When attempting to expand the Catalogs in the linked server object defined,  I received below message.How can I fix this problem?

The EXECUTE permission was denied on the object ‘xp_prop_oledb_provider’, database ‘mssqlsystemresource’

Answer: The overall fix is to grant execute rights to the logon requiring the privileges on SYS.XP_PROP_OLEDB_PROVIDER.

Step 1 >>  Map the user in the master database

USE [master]
GO
CREATE USER [myuser] FOR LOGIN [myuser]

Step 2 >> GRANT EXECUTE ON SYS.XP_PROP_OLEDB_PROVIDER

GRANT EXECUTE ON SYS.XP_PROP_OLEDB_PROVIDER TO [myuser]

These are the basic steps. But quite often in an organizations SQL Server Security Policy   , direct execute rights on an objects are prohibited.  Normally a database role   may be required

Read more on sql server linked servers 

SQL Server Linked Servers and User permissions