Learn Romanian: M-am trezit târziu azi-dimineață.

In Romanian, waking up is an action you do to yourself. The verb is a se trezi. If you drop the reflexive pronoun and just say am trezit, it means you woke someone else up, and the listener will expect to hear who (for example, l-am trezit, meaning 'I woke him up').

What exactly is contracting to form m-am?

It is a contraction of the reflexive pronoun mă (myself) and the past tense helper verb am (I have). When spoken at normal speed, the ă in mă is dropped to avoid having two vowels back-to-back, combining them into m-am.

Why is azi-dimineață written with a hyphen?

It is a fixed compound word acting as an adverb, literally meaning 'today-morning'. Romanian frequently uses these hyphenated compounds for specific times of day. While you could technically say în această dimineață (in this morning), azi-dimineață is the standard, everyday way to express it.

Can I change the word order to M-am trezit azi-dimineață târziu?

You can, but it changes the emphasis. Placing târziu (late) immediately after the verb makes your lateness the core piece of information. Moving it to the end is grammatically fine but sounds a bit less natural in casual conversation.