Learn Dutch: Omdat de verzekering betaalt, ben ik blij. - Because the insurance pays, I am happy.

This is because of inversion. In Dutch, the main verb of a sentence usually needs to be in the second position. Since the entire first clause (Omdat de verzekering betaalt) counts as the first piece of information, the main verb (ben) must come immediately after the comma. This pushes the subject (ik) to the other side of the verb.

If I added more words to the first part, where would they go?

They would go before the verb. The word omdat introduces a subordinate clause, which forces the verb to the very end of that clause. For example, if you wanted to say 'Because the insurance pays everything', you would put the object before the verb: Omdat de verzekering alles betaalt, ben ik blij.

Could I use the word 'want' instead of 'omdat' here?

No, you cannot start a sentence with want. Both want and omdat mean 'because', but want only connects two main clauses and must go in the middle. If you wanted to use want, you would have to flip the ideas: Ik ben blij, want de verzekering betaalt.

What happens to the word order if I flip the two halves of the sentence?

If you start with the main clause, it follows normal word order: Ik ben blij. The omdat clause still keeps its verb at the end, so the full sentence becomes: Ik ben blij omdat de verzekering betaalt. Notice that because you didn't start the sentence with a subordinate clause, there is no inversion for 'ik ben'.