First of all, the quote is from this chapter of the Kishkindha Kanda of the Ramayana, where Vali accuses Rama of acting sinfully in killing him:
My skin is unwearable, holy people forbid my hair and bones, and uneatable is my meat for your kind of reputable people. Raghava, five kinds of five-nailed animals, viz., a kind of wild rodent, a kind of wild-boar, a kind of lizard, a hare and fifthly the turtle are edible for Brahmans and Kshatriyas. Sensible people will not touch my skin and bones, oh, king, nor meats from my body are to be eaten, such as I am, a five-nailed animal, I am killed.
As I discuss in this answer, Rama gives a thorough defense of his actions, and at the end of it Vali agrees Rama was in the right. In any case, Vali is referencing the rule given in this chapter of the Manu Smriti, which lists various foods that are and are not permitted for Dvijas (members of the first three castes):
Let him not eat solitary or unknown beasts and birds, though they may fall under (the categories of) eatable (creatures), nor any five-toed (animals).
The porcupine, the hedgehog, the iguana, the rhinoceros, the tortoise, and the hare they declare to be eatable; likewise those (domestic animals) that have teeth in one jaw only, excepting camels.
Verse 18 lists exceptions to the general prohibition given in verse 17.
In any case, even if the Manu Smriti doesn't forbid the eating of certain animals by Dvijas, there are still excellent reasons to be a vegetarian. It gives you good gunas that can lead you to be a more virtuous person, it's more compassionate to other living beings, and as the Manu Smriti itself says, abstaining from meat generates as much Punya as the performance of a hundred Ashwamedha Yagnas. So go vegetarian!