Do Spiritual Masters, Yogis, Avatars and Demons of the past transcend the veil of death and appear in the here and now of this phenomenal world and if so how do they do it and where have they come from?
By the power of Yoga a Yogi can transcend birth and death. He is then like the Jivanmukta (liberated while living).
It is not necessary for him to take birth then but if he chooses to do so he can continue to live in whatever form and in whatever world. It's his free will. So, he can again and again appear in this world or for that matter any worlds whenever he wishes to.
So, the answer to "how they do it" is by practicing Yoga.
Quoting from the Yoga Tattva Upanishad:
- Then his body becomes strong and he does not know death. That great-minded man does not die even during the deluge of Brahma.
104-105. Then he should practise Dharana for a period of six Ghatikas (2 hours, 24 minutes). Restraining the breath in (the region of) Akasa and contemplating on the deity who grants his wishes - this is said to be Saguna Dhyana capable of giving (the Siddhis) Anima, etc. One who is engaged in Nirguna Dhyana attains the stage of Samadhi.
Within twelve days at least, he attains the stage of Samadhi. Restraining his breath, the wise one becomes an emancipated person.
Samadhi is that state in which the Jivatman (lower self) and the Paramatman (higher Self) are differenceless (or of equal state). If he desires to lay aside his body, he can do so.
108-109(a). He will become absorbed in Parabrahman and does not require Utkranti (going out or up). But if he does not so desire and if his body is dear to him, he lives in all the worlds possessing the Siddhis of Anima, etc. 109(b)-110. Sometimes he becomes a Deva and lives honoured in Svarga; or he becomes a man or an Yaksha through his will. He can also take the form of a lion, tiger, elephant, or horse through his own will.
This is possible though for the most adept Yogis and not for the ordinary ones.