How do I know that my bitcoins are real
Enter your Bitcoin address (the one you were paid into) into a public blockchain browser such as https://blockchain.info/. That will show you the relevant entries in the global shared Bitcoin ledger, which confirms that you, as the holder of the private key for that address, are the owner of those funds.

If you lose control of your private key for any reason, then your bitcoins might not be yours for much longer.