This article also has some relevance to the Ramayana story, but it is more than that as you will see.

moehana raagam, Adi tALaM

moehana raama, mukha jita soemaa
mudduga balkumaa, jagan //moehana

moehana raamaa, modaTi daivamaa,
moehamu neepai molaci yunnadiraa //moehana

dhara manujaavataara mahima vini
sura kinnara kimpurusha vidyaadhara
surapati vidhi vibhAkara candraadulu
karagucu praematoe
vara mRuga pakshi vaanara tanuvulacae
girini velayu seetaavara cira kaalamu
guri tappaka maimaraci saevimciri
vara tyaaga raaja varada akhila jagan //moehana

Meaning and summary in English:
moehana raama = oh enchanting Rama; mukha jita soemaa = with face rivaling the moon;
mudduga balkumaa = speak sweetly (to us); jagan-moehana Rama = world-enchanting Rama!

moehana raamaa = oh enchanting Rama; modaTi daivamaa = Oh very first (supreme) Lord;
moehamu = love, aspiration; neepai = upon you; molaci yunnadi = has sprouted (has become
centered); raa = a term of familiarity and endearment.

dhara = on earth ; manujaavataara = human incarnation ; mahima = miracle; vini = having heard
sura = gods ; kinnara kimpurusha vidyaadhara = various celestial beings ;
surapati = indra ; vidhi = brahma ; vibhAkara = the Sun ; candraadulu = the Moon and others ;
karagucu = melting ; praematoe = with love ;
vara = blessed ; mRuga = animal ; pakshi = bird ; vaanara = ape-men; tanuvulacae = in those
forms;
girini velayu = residing on the hill ; seetaavara = oh beloved of Sita ; cira kaalamu = for a
long time;
guri tappaka = without missing aim (with unwavering concentration); maimaraci = forgetting
themselves; saevimciri = they worshipped you;
vara = blessed ; tyaaga raaja varada = Tyagaraja's savior; akhila jagan moehana Rama = Oh,
the entire-world- enchanting Rama!

Oh enchanting Rama, with a face rivaling the moon, speak sweetly to us.
Oh Supreme Lord, my devotion has been firmly centered on you!
When they heard of your miraculous appearance in human form on this earth, Brahma, Indra and
such Gods and other celestial beings were so charmed by their love for you, that they too
came to earth in the guises of blessed animals, birds and vaanaras - just so that they can
keep on worshipping you for a long time with unwavering attention, Oh Sita's beloved, Oh
blessed Tyagaraja's savior, Oh charmer of the entire world, Oh Rama!

Some personal notes:
The opening phrase epitomizes and sets the mood for the rest of the kruti - Oh enchanting Rama! - moehana means, charming, enchanting, exquisite loveliness. IF you remember, Vishnu's sole female incarnation was called "moehini" - because she charmed everyone with her grace and beauty. In the case of Rama, Valmiki, the first poet himself described him thus, "pumsA moehana roopAya". Tyagaraja indicates all this and more with just that opening phrase.

Why did Rama incarnation happen? Ravana was persecuting gods and various celestial beings. They all went to Vishnu and pleaded with him to do something about it, so Vishnu agreed to go down in human form to take care of the business (Ravana had a boon that he won't be killed by any of the celestial beings). Anyway, since Vishnu was taking on the human incarnation for their sake, all these celestial plaintiffs felt it their duty to come and help him as they could - so they took the forms of various animals, birds and the vaanaras all of which played a unique role in the rest of the drama. - this is according to adhyatma Ramayana. Tyagaraja puts a slightly different and more delightful spin on this: Yes, all these gods came down in the forms, but why? Because Rama was so enchanting - they couldn't bear to be away from him.
By taking these humble forms, they are obtaining the opportunity to be close to him, to continue worshipping him for longer time (i.e., for the duration of his human incarnation) with unwavering attention. That's why they all came down! Not because Rama really needed their help.

All the above was in the last stanza. Before going on to that, Tyagaraja offers a sub-theme in the second stanza; if the power of his enchantment is so great on these celestial beings, then how can Tyagaraja, a mere humble mortal, who is already a devotee, resist it? So it is no surprise that Tyagaraja's devotion is firmly planted in Rama. The word used by Tyagaraja to denote this - "molaci" - means sprout or take root illustrates Tyagaraja's facility in employing the mot juste .

I absolutely love this song. The lyrics, the music and the mood are just heavenly. It sounds as if Tyagaraja captured the divine grace of Rama's incarnation, blended it with the soul of the moehana raagam, and squeezed the combined essence into this song. I have heard just the opening line being sung/played 10-20 different sangatis - it's just marvellous. Tha soaring notes in the anupallavi at "modaTi daivamA" and at the end of the caraNam at "maimaraci saeviMciri" create an ethereal sensation. I have one gripe though - most vocalists, even those who know Telugu well, murder the lines at the end - "guri tappaka mai maraci saeviMciri" - they always sing it as "gurita paaka mai maracisae viMciri". Sigh!