Why in the morning do we say מודה אני and not אני מודה?
In Biblical Hebrew the verb is often placed before the subject. (See, for example, almost all uses of vayomer and vaydaber in Torah.) This is different from Modern Hebrew, where the subject is often placed before the verb. (In fact, I've gotten funny looks when attempting to use the Biblical style in conversations.)
מודה אני isn't unusual in this regard. We also say מודים אנחנו לך, for example, and ברוך אתה in lots of places. The prayers are, by and large, written in the style of Biblical (and rabbinic) Hebrew and preserved in that form.
DonielF added the following in a comment:
The idiom אני מודה, according to Sefaria’s search engine, appears 4 times across Gemara and Midrash (only one of them is in the Gemara), while מודה אני appears 30 times across the same sources, including a broader range of Midrashim, plus Tosefta and Avos d’Rebbi Nassan. So on the one hand, it’s not unheard of to use אני מודה, but on the other hand, it’s also significantly more common to use מודה אני.
I remember reading an article(maybe Aish.com?) that explained the reason for starting our day with the word "modeh" and not "ani": so that we don't start our day with "ani"/me but, rather, "modeh"/I am thankful.
The reason is very deep that I think can sum up our destiny, based on how we approach life. If you start your day with "ani"/me, then your life depends on you(which could be a curse). But if you start your day with modeh, then you start it with gratitude(and humility).
Too many times can we feel frustrated if things don't go our way. But why? Because we think it depends on "ani"/me. Emunah teaches us the opposite. Since HaShem is active in our lives and everything He does is for our best, then I am "modeh"/I am thankful.
To further elaborate, a person who takes the approach of "modeh"/ I am thankful, has the path set towards olam haba because they submit to God's control and nullify themselves to the will of God. This is unlike the one who takes the approach of "ani"/me.
Monica's answer (and the supporting comments) is probably the correct one. I would just add that, otherwise, "fronting" a constituent is a means of emphasis.
Biblical Hebrew is usually VSO (which means Verb Subject Object), so the verb should lead, and bringing the subject or object to the beginning is a way of emphasizing it. However, if the typical order for the present tense were to be to put the subject (ani) first, then the purpose would be to emphasize the gratitude.
See here, in Modality and the Biblical Hebrew Infinitive Absolute.