Much of it happened in the Islamic world and then filtered into Latin Europe, mostly via translators in Spain. Greek scientific and philosophical writing had been preserved in the eastern provinces of the Roman Empire much more than the western - Egypt, Syria, etc. - and the Muslims inherited this when they conquered these areas in the seventh century. By the ninth century they had translated most of it into Arabic and began to commentate and continue the development of this tradition. They also adopted things like a decimal place-numeral system from the Indians (hence our "Arabic," as opposed to "Roman," numerals) which made mathematical calculations faster and more accurate. Most mathematical and technical terms in English that begin with "al-" (Arabic for "the") come from Muslim thinkers. Algebra, Algorithms, Alchemy (the beginnings of chemistry), etc. Thinkers like Ibn Sina ("Avicenna" to westerners) both commentated on the works of Aristotle but made original advances in fields like math, astronomy and medicine. Avicenna, working in medicine, pioneered scientific methods in using systematic experimentation to get the best results. Muslim astronomers even probed the idea that the earth revolved around the sun, although none of them finally adopted the idea.
All of this started to filter into Europe, with some early translations in the 10th-11th centuries and a big wave in the 13th. Roger Bacon in the 13th century asserted that observation and experience were surer guides to knowledge than the authorities or pure logic relied on by most medieval thinkers. Another Bacon, Francis (not sure if there's a relation), would develop this into full-on empirical method in the 16th century. Meanwhile, medieval Europe had become very adept at borrowing technologies developed elsewhere (like gunpowder or paper) or devising its own (like the magnetic compass) and putting them to practical use. This would lead to things like the telescope which made the discoveries of Galileo possible
