In the Ramayana, Shri Ram was an avatar of Vishnu; in the Gita or Mahabharata, Shri Krishna was an avatar of Vishnu. Everything they did in human form, they were perfectly able to do it in their original/God form.
Then why did they opt for a human avatar/form? Was it in order to come and live amongst normal people and then help them?
This is indeed a very good question. But unfortunately no specific reason can be pointed out as the sole cause for God's incarnations:
हरि अवतार हेतु जेहि होई। इदमित्थं कहि जाइ न सोई।। [Ramcharit Manas - 1.121]
However, from scriptures and whatever sages have said three primary reasons can be pointed out,
1. Establishment of Dharma 2. Deliverance of saintly people 3. Destruction of the wicked people
Shri Krishna has Himself told this to Arjuna as we know from the Gita:
paritranaya sadhunam vinasaya ca duskrtam
dharma-samsthapanarthaya sambhavami yuge yuge [BG - 4.8]
The same reason has also been stated for Rama:
जब जब होइ धरम कै हानी। बाढहिं असुर अधम अभिमानी।।
करहिं अनीति जाइ नहिं बरनी। सीदहिं बिप्र धेनु सुर धरनी।।
तब तब प्रभु धरि बिबिध सरीरा। हरहि कृपानिधि सज्जन पीरा।।
असुर मारि थापहिं सुरन्ह राखहिं निज श्रुति सेतु।
जग बिस्तारहिं बिसद जस राम जन्म कर हेतु।। [Ramcharit Manas - 1.121]
Meaning
Whenever dharma declines and the demons, wicked and the haughty increase,
when they do such immoral work that they can not be described, and when Brahman, cow, gods, and earth are in danger,
In those times God taking different kinds of bodies take away the pain and suffering of the saintly persons,
Killing the demons He reinstates the gods and preserves the order of His own Vedas,
He spreads His glory throughout the world, for Rama's birth know these are the cause.
But then the question is why the almighty God, by whose mere thinking thousands and thousands of universes get annihilated, would take birth to kill mere demons and wicked humans who are just part of the universe itself? Can't He just do it by His will without taking so much trouble of taking birth here in this mortal world?
Hence, the above reason alone is not satisfactory. So Shrimad Bhagavatam states an even more appropriate reason for the cause of God's descend as a human. God takes avatar as a human not just to kill the demons but to teach us the way of life by practically observing it Himself:
martyāvatāras tv iha martya-śikṣaṇaṁ
rakṣo-vadhāyaiva na kevalaṁ vibhoḥ [SB 5.19.5]
This reason seems sensible as from Shri Rams's life we learn how to be a good son, good brother, good king, etc. But then the question arises, what is the necessity of God to take avatar and teach us Himself when prophets, saints and sages already do the job of teaching by living saintly life themselves? Again, if we take Shri Krishna's birth as example, then there are also many activities by Him which we should never try to do it ourselves. Hence, this can't be the sole reason. So Bhagavatam says, "However we may argue, there can be no other reason than amusement for which He who has no birth takes birth here":
na te ’bhavasyeśa bhavasya kāraṇaṁ
vinā vinodaṁ bata tarkayāmahe [SB - 10.2.39]
That is, God takes birth here for His amusement or entertainment. But this can not be the actual reason because He who Himself is always blissful (sadchidanda) would require no other thing for His amusement. Only they who are not happy, tired or stressed require and depend upon other things for happiness and amusement. But God being always blissful why would He require anything else for His pleasure or amusement? Hence, Bhagavatam states the most appropriate reason for God's descend is to bestow the highest benefit upon humankind by manifesting His pastime here:
nṛṇāṁ niḥśreyasārthāya
vyaktir bhagavato nṛpa [SB - 10.29.14]
This reason seems sensible because without the incarnations of God, the world would be bereft of His sweet pastimes. Without which there would be no scriptures like Ramayana, Bhagavata, etc. and the devotees would have nothing to think and talk about. Only the concept of avyakta (unmanifested) nirakara (formless) God would have remained which is hard to comprehend and practice. So one of the primary reasons for God's descend as a human is for bestowing the mercy of devotion to people and even to the transcendentalists who worship the unmanifested formless absolute (avyakta nirākara brahma):
tathā paramahaṁsānāṁ munīnām amalātmanām
bhakti-yoga-vidhānārthaṁ kathaṁ paśyema hi striyaḥ [SB - 1.8.20]
As for the why take an avatar part, it's in the most popular verse of the Bhagavad Gita:
yada yada hi dharmasya
glanir bhavati bharata
abhyutthanam adharmasya
tadatmanam srjamy ahamWhenever and wherever a decline of righteousness and predominance of unrighteousness prevails, at that time I manifest myself personally, O descendant of Bharata.[1]
For your second part:
Why gods had to take human avatar or take a human form in order to come and live amongst normal people?
Avatars are not always of human form. The first 4 avatars (Matsya, Kurma, Varaha, Narasimha) were animals in the order of aquatic animal, amphibious animal, and land animals, before taking human form.
This in a way supports the idea of the evolution of consciousness (as well as species) and instilled values to emulate for the consciousness (and species) present at that time.
And why human form?
To prove to the humans that the creator of laws of the universe can operate within the laws and that living a righteous life within this framework is humanly possible.
"The avatar creates the capacity in consciousness for every human to achieve a similar victory of realization."[2]
On a side note, an avatar is significantly different from a prophet in that every human can attain the consciousness of an avatar while a prophet is a "chosen" special person whose consciousness cannot be attained by humans.
[1] http://www.bhagavad-gita.org/Gita/verse-04-07.html
[2] Malothra R., Being Different: An Indian Challenge to Western Universalism, Chapter 5
Vishnu's avatars typically descend for a very specific purpose. An oft-quoted passage from the Bhagavad Gita describes the typical role of an avatar of Vishnu—as bringing dharma, or righteousness, back to the social and cosmic order:
" Whenever righteousness wanes and unrighteousness increases I send myself forth. For the protection of the good and for the destruction of evil, and for the establishment of righteousness, I come into being age after age. (Gita:4.7–8)"
(Source:Wikipedia)
And if you go with the story of Avatars of Vishnu or devi, then you know this Avatar are to help Humans because of vardan's of Asura's which cause trouble for all. For example Hiranyakashyap can't be killed by deva or humans or animals or asura. So need of Narasimha avatar came whcih is none of them from the vardan.
Avatar's also needed to guide us in a right way. Rama is associated with Dharma teahcing to humanity. Similarly Krishna is associated with teaching us about Karma and given us Geeta.
Historically avatars like Rama and Krishna came to wage wars against the ungodly. Avatars can also come to spread spiritual message by directly showing people how to live a stainless life. Let me post some direct quotes from Sri Ramakrishna:
"God can be directly perceived in a man in a tangible form. Seeing an Incarnation of God is the same as seeing God Himself. God is born on earth as man in every age." (March 9, 1883)
"God has different forms and He sports in different ways. He sports as Isvara, deva, man and the universe. In every age He descends to earth in human form, as an Incarnation, to teach people love and devotion. There is the instance of Chaitanya. One can taste devotion and love of God only through His Incarnations. Infinite are the ways of God's play but what I need is love and devotion. I want only the milk. The milk comes through the udder of the cow. The Incarnation is the udder." (June 25, 1883)
"The Incarnation of God is a part of the lila of Sakti. The purpose of the Divine Incarnation is to teach man ecstatic love for God. The Incarnation is like the udder of a cow, the only place milk is to be got. God incarnates Himself as a man. There is a great accumulation of divinity in an Incarnation, like the accumulation of fish in a deep hollow in a lake." (September 7, 1883)
REF: All the quotes are from the English translation of Ramakrishna Kathamrita]
