Which scriptures say that the Kings and Kshatriyas are allowed to hunt animals?
Bhishma says this to Yudhishthira in Mahabharata Book 13 of Anushasana parva Chapter CXVI http://www.sacred-texts.com/hin/m13/m13b081.htm
"Bhishma said, 'It is even so, O mighty-armed one, as thou sayest. There is nothing on earth that is superior to flesh in point of taste. There is nothing that is more beneficial then flesh to persons that are lean, or weak, or afflicted with disease, or addicted to sexual congress or exhausted with travel. Flesh speedily increases strength. It produces great development. There is no food, O scorcher of foes, that is superior to flesh. But, O delighter of the Kurus, the merits are great that attach to men that abstain from it. Listen to me as I discourse to thee on it. That man who wished to increase his own flesh by the flesh of another living creature is such that there is none meaner and more cruel than he. In this world there is nothing that is dearer to a creature than his life. Hence (instead of taking that valuable possession), one should show compassion to the lives of others as one does to one's own life. Without doubt, O son, flesh has its origin in the vital seed. There is great demerit attaching to its eating, as, indeed, there is merit in abstaining from it. One does not, however, incur any fault by eating flesh sanctified according to the ordinances of the Vedas. The audition is heard that animals were created for sacrifice. They who eat flesh in any other way are said to follow the Rakshasa practice. Listen to me as I tell thee what the ordinance is that has been laid down for the Kshatriyas. They do not incur any fault by eating flesh that has been acquired by expenditure of prowess. All deer of the wilderness were dedicated to the deities and the Pitris in days of old, O king, by Agastya. Hence, the hunting of deer is not censured. There can be no hunting without risk of one's own life. There is equality of risk between the slayer and the slain. Either the animal is killed or it kills the hunter. Hence, O Bharata, even royal sages betake themselves to the practice of hunting. By such conduct they do not become stained with sin. Indeed, the practice is not regarded as sinful. There is nothing, O delighter of the Kurus, that is equal in point of merit, either here or hereafter, to the practice of compassion to all living creatures
So Kshatriyas are allowed to hunt all wild deer species so as to eat their meat which they acquire at risk to their lives.
- Hunting of animals whose meat you don't plan to eat would not be allowed.
- Hunting should involve risk to hunter's life, so hunting using traps, dogs and modern rifles etc which keep hunters safe would not be allowed.
Justification for this act are twofold. 1. All wild deer have been dedicated to devas and pitris in ancient times, so hunting them is same as sacrificing them in vedic ritual as far as its sinfulness is concerned. 2. Risk taking during hunting tests kshatriya's prowness. This is probably most close to battle situation kshatriya will come without actually doing battle and killing other humans. Hunting would test and train Kshatriya's courage , athletic abilities and ability to think on one's feet. This was probably best way to stay sharp and trained during peace times.
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