I may have found an answer here: https://devcentral.f5.com/questions/difference-between-root-cert-intermediate-cert-and-ssl-cert

> One more thing one the public key of your server certificate. There is
> the same mathematical relationship as described above between the
> public key and private key of your server certificate. 
> 
> It will be used during the SSL handshake. The client encrypts some key
> material servers public key (retrieved by the client out of the server
> certificate) and sends it to the server. It can be decrypted by the
> server using the private key. If the server does not have the matching
> private key it cannot decode the key material and the handshake
> brakes.