Learn Dutch: Vandaag moet ik vroeg opstaan. - Today I have to get up early.

Because the sentence starts with the time word vandaag (today). In Dutch, the verb always takes the second position. When you put a word like vandaag in the first position, the verb moet has to come next, pushing the subject ik (I) to the third position.

How does the modal verb moeten work in this sentence?

Moet is the ik (I) form of the verb moeten, which means to have to or must. It shows obligation. When you use a modal verb, it is the only verb in the sentence that changes its form to match the subject.

Why is opstaan (to get up) at the very end of the sentence?

When you use a modal verb like moeten, the main action verb gets pushed to the very end of the sentence. This main verb always stays in its full, dictionary form (the infinitive), which is opstaan here.

Isn't opstaan a separable verb? Why didn't we split it?

You are right that opstaan is a separable verb! In a simple sentence you would split it, for example: Ik sta vroeg op (I get up early). But because we used a modal verb here, opstaan is pushed to the end as a full infinitive, so it stays glued together.