How does DH work when combined with ECC or RSA? I know that Diffie-Hellman is used to create keys in a secure way over an insecure channel. But there is one thing which I cannot understand: I see that a lot of sites use ECC or RSA alongside DH.

But why would you need to implement ECC or RSA if you already have a shared secret or key with each other? Can't you already use to strong symmetrical encryption?
What's missing is the authentication of the entities. If you don't authenticate the entity then you don't know who you've established the master secret with. This means an attacker can pose as a man in the middle or the attacker can simply act as one of the entities.

You can use static DH key pairs, but in that case the DH public keys must be trusted and the scheme loses the forward secrecy property.

In TLS 1.3 only DHE and ECDHE will be used for key agreement, which means that the key pairs are fully ephemeral and the forward secrecy property is indeed in effect.