Vedic Math has always been a mystery. As most things useful, like yoga, it started getting noticed and appreciated in India, only when the Western culture started openly hailing it as India's greatest contribution to civilization.
When the Vedas were written, a part of it was the Ganita Sutras, which contained mathematical deductions and all the information which today forms Vedic Math. After centuries of being buried in oblivion, they were discovered in the beginning of the 20th century, at a time when there was a great and rising interest in ancient Sanskrit texts, especially in Europe. The Ganita Sutras even though discovered, were ignored as not much traditional and obvious forms of Math can be found in them.
At the beginning of the 20th century, between the years 1911 and 1918, Sri Bharati Krishna Tirthaji, a scholar of several subjects such as Sanskrit, Mathematics, History and Philosophy rediscovered Vedic Math in the Ganita Sutras. After several years of extensive study, he managed to put together 16 volumes of work, which included Sutras, which were a string of Mathematical formulae thus pioneering Vedic Mathematics in 1965. Sadly, these 16 volumes were lost soon after. He then put together a single volume, which included all the main formulae and sutras which define Vedic Math today. This volume was published five years after his death.
A copy of this volume reached London in the late 1960s and was immediately hailed as a new alternative system of Mathematics. Renowned British Mathematicians such as Kenneth Williams, Andrew Nicholas and Jeremy Pickles took interest in this volume on Vedic Mathematics, extended the work of Sri Bharati Krishna Trithaji in this volume and started delivering lectures on the same in London. In 1981, the introductory volume along with the extensions by these renowned mathematicians was collated into a book titled Introductory Lectures on Vedic Mathematics. Mathematician Andrew Nicholas made several trips to India between 1981 and 1987 and renewed the concept and subject of Vedic Math among scholars and teachers in India. This is when the journey of Vedic Math truly started in the land where it was invented decades ago.
These sutras form the basis of what Vedic Math means to us today.