Could anyone please help me understand MSSQL Compatibility Level?
Maybe Compatibility_level = 90 is of MSSQL-2005, then is it really been upgraded to MSSQL-Server-2012 or not? how may I know?


If it's running on the 2012 instance, then that database's metadata has been upgraded. So you're good.


  Could anyone please explain me why restored database also showing Compatibility_Level as '90' and not '110'?


When you restore a database onto a newer version (called an upgrade), it's compatibility level is set to the lowest supported version for that instance.  In this case, 90 is still functional on the 2012 instance, so it was left as-is.


  Also, If it is still contains featured of MSSQL-Server-2005 only, then what is the better way to upgrade it to MSSQL-Server-2012?
  
  Upgrade is as much simple as "ALTER DB"?? like


Yes, it really is just as easy as ALTER DATABASE ... SET COMPATIBILITY_LEVEL.

You can remain at the lower compatibility level as long as you like.  However, to use newer features introduced since version 90, you'd need to update to the later compatibility levels.  You can see a list of some of the changes here: 

Differences Between Lower Compatibility Levels and Levels 110 and 120

Also note that, starting with SQL Server 2014 (level 120), the 90 compatibility level is not available.  So the next time you upgrade, your database will automatically be updated to compatibility level 100.  You can see a table of supported levels on each engine version here:

ALTER DATABASE (Transact-SQL) Compatibility Level - Arguments