By becoming your own Certificate Authority, you provide authenticity to the certificate by creating a signature of it. This is done via a signature which is signing via the private key. The public key is then used to verify that the signature was indeed signed by the private key.
Because only the CA (you) knows the private, you trust that the CA verified the certificate and it is authenticate. To trust the CA, you install the CA's certificate (which contains the public key) into your ring of trust, generally in Windows this is done by double clicking the certificate file and verifying you wish to trust it. In Firefox it's done by adding it within the Certificates area of the Options.
The key to the above is making sure that no other CA in your ring of trust could potentially sign the signature. This is why the CAs verify the owner of the domain before signing a certificate for that domain.
If you'd like to learn more of the details of cryptography including public/private:
An amazing book on the subject is Applied Cryptography.
A much simpler book is Cryptography Decrypted.