How did the Buddha become the Buddha?

Buddha means the Fully Enlightened One. He became the Buddha through the realisation of the intrinsic / true nature of all things in the universe, including existence / mind & body / life.

The historical Buddha was born in the Himalayas (Nepal, year 623 BC), as Prince Siddhartha of the Sakya Kingdom. The natives of ancient Nepal were the Kirat people / Kiratis (better known today as the Gurkhas), comprising of Tamang, Gurung, Sherpa, Rai etc.

At age 29 years old, Prince Siddhartha left Nepal in search of a way to end rebirth, old age, disease and death. After 6 years of spiritual pursuit, at age 35 years old, he realised the Four Noble Truths:

1) Life is Dukkha (i.e. worldly existence is insatisfactory as all conditioned things are subjected to change, including self, heavenly existence and even the sun)
2) Cause of Dukkha (due to ignorance of point 1, which leads to craving, grasping and therefore, endless rebirth)
3) End of Dukkha (by getting rid of ignorance and therefore, grasping)
4) Way to end of Dukkha (by practising the Noble Eightfold Path which leads to Enlightenment and therefore, Nibbana)

Nibbana is the ending of rebirth (and therefore, decay and death) and state of Highest Happiness. The Buddha describes Nibbana as Beyond the Worldly (beyond space-time continuum), Unique, Not Existence Nor Extinction, Freedom and Peace.