Learn Dutch: Doordat hij hard rijdt, gebeurt er een ongeluk. - Because he drives fast, an accident happens.

This is the rule of inversion! In Dutch, the main verb must be the second 'block' of information in a main clause. Because the entire subordinate clause (Doordat hij hard rijdt) acts as the first block, the main verb (gebeurt) must come immediately after it to take the second position.

Why is the verb rijdt at the very end of the first part?

The word doordat introduces a subordinate clause. In Dutch subordinate clauses, the conjugated verb is pushed to the end. That is why it is hij hard rijdt instead of the normal main clause order hij rijdt hard.

Could I use omdat instead of doordat to say 'because'?

You could, but there is a subtle difference. Omdat usually introduces a reason, often involving a human choice. Doordat describes a strict cause and effect. Because the fast driving is the direct physical cause of the accident, doordat is the most precise word to use here.

What is the word er doing in the phrase gebeurt er een ongeluk?

The word er acts like 'there' in the English concept of 'there happens an accident'. In Dutch, when a subject is indefinite (like een ongeluk, meaning 'an' accident instead of 'the' accident), you typically must use er alongside the verb.