I want to know how Anonymous bitcoin is
Bitcoin itself is not anonymous at all.
Given the proper precautions, bitcoin can be used in an anonymous way, but it requires significant effort and understanding of the methods that you expose your identity when performing transactions. The amount of effort and understanding required depends on who it is you are trying to hide from.
I know it's impossible to get your name, address, age & all your private information.
Not necessarily. That depends on who is looking, and what efforts you've made to hide your identity.
but still people will be able to see all with your address (your transactions & moves) on blockchain.info, so is it possible to make off chain transactions?
Yes. If you are willing to trust other services, then it is possible to make off-chain transactions or to obfuscate your transactions with mixers or coinJoin.
I mean make a transaction without making it show on blockchain.info or blockexplorer or any similar websites?
Yes, but the method depends on what you are trying to accomplish. You can obfuscate the transaction with CoinJoin or you can trust a third party service to facilitate off-chain transaction.