It is unknown how Morgoth accomplished this.
Some say he did it long ago, when he still walked Arda in earthly shape, during the Late Wars of Beleriand and when the Noldor came to challenge him. It is said that in Angband he held many slaves and captives from the race of Men and that he performed terrible and wicked experiments on them, similar, though not identical, to the ones he performed on Orcs. His means in doing so are unknown, but what is important is that he succeeded, changing the very bodies and blood of the captured Men. Weird and unnatural things started happening with them and around them. They all showcased very unique gifts and were stronger, more resilient and often more intelligent compared to other Men. Because of this, he used them in hardest tasks, making them dig pits and make dungeons for his armies. Sauron, Balrogs and others of his greatest servants were the only daring to approach these slaves, because Orcs feared them and made no contact with them. He allowed these Men only to procreate with each other.
Then, one day, to surprise of everyone, he released these Men from Angband, letting them, and their children, go back to the lands of their forefathers, and some of them even joined the Three Houses of Edain in wars against him.
Why did he do this?
Many believed that this was a reflection of his former greatness, a sparkle of goodness that he once held as mightiest among the Valar and that he did this out of pity, feeling great shame and temporary repentance. But those who were wiser did not believe this to be the case and knew that, behind this act of "mercy", Morgoth was hiding a wicked plan, a plan of horrible and utmost treachery that would be revealed in the Last Days when the Sun fails and the Moon is dead.
After he was banished into the Void, Morgoth could no longer reach Arda in any way except small thoughts and actions. Though all of Arda contained a part of Morgoth ingredient, he had no direct control over what happen upon it. Instead, Sauron, who took up his master's mantle as the Enemy of Free Peoples, became the second Dark Lord and he remained an echo of Morgoth in later ages, speaking with his voice and seeing with his eyes.
Now Sauron knew about the members of the race of Men that his master released from Angband and he sought their descendants. He would go after them through every land and chase them like animals, capturing them and bringing them to his fortresses and dungeons. There, he would continue his master's work, torturing and changing these Men, turning them into something that looked like the race of Men, but was also superior to it. Then he would let them go back to where they came from, and these Men would procreate with the rest of their race without any problems or complications. Soon, the first of these true Mutants was born: Sauron himself gave her a name, calling her Selene.
What was Morgoth's plan here? Why did he not instead torture these Men and turned them into monsters like Orcs, blind servants of his will, never able to resist him?
Because his thoughts and plans went beyond the Wars for the Jewels and, unlike what the Valar thought, he knew that he will not win those wars, instead knowing that he will one day return, when the Doors of Night would be broken, when he would destroy the Sun and Moon and his power would be restored and the Last Battle would begin. So he took these Men and changed their inner body instead of their minds and form like he did with Orcs. Through slow process of corruption and nurturing, his corruption grew in them as a mutation, giving them unique gifts that went far beyond anything that Men or even the Elves could do, powers that could even challenge the Rulers of Arda. And inside of them, he implanted a secret, a secret sparkle that would run through all of them, whether they knew it or not.
For he knew that other, powerless Men, would fear these gifts and that they would be instead despised as curses. Mutants were to be betrayed and disowned by their own families and loved ones, until, after a long struggle, the rift between the two could finally be erased and Mutants and Men could live in peace, even love.
And Morgoth counted that he would return very soon after this peace began and that, when he would descend for Dagor Dagorath and summoned them, they would answer his call, even against their own will. And in the Final Battle, Mutants would be fighting against all their friends, familes and loved ones in his name, either forgetting who they were or converting to his side or even working like mindless animals under his command. Some would be able to resist, no doubt, and not all would enter his service, but all Mutants would be affected by this and thus the unwilling treachery of mankind by the side od Mutants would take place.
And that is why this, along with the torture and corruption of Orcs, was the vilest, cruelest and most horrible deed of Morgoth and the most hateful to Ilúvatar.