An idea, that to know this world is to also know God, developed as a result of new thinking. And because God and nature were being reevaluated, the garden acted as an experimental space for both changed relationships. Along with humanism, the Renaissance saw a passionate rediscovery of Ancient Greek and Roman literature. It was examined, translated, studied, and elements began appearing in painting and sculpture of the time. Just like art, gardens saw an interesting blend of religious art, both Christian and Classical. Gardens very often had allegorical themes, which utilized a variety of characters to represent everything from ancient myths to the owner of the garden's occupation and goals in life, as can be seen throughout Cardinal Ippolito's Villa d'Este. The picture below shows the garden's Fontana Della Rometta, which represents the Cardinal's desire to move to Rome and become the Pope.
The renewed interest in Ancient Roman life brought about the desire to emulate it as much as possible. As such, countryside villas like Villa d'Este were a reincarnation of Ancient Roman homes. Compared to gardens of Medieval times, the Renaissance gardens were physical representations of the influences of Renaissance thought. They were expansive to reflect expansions in thinking, outwardly directed in comparison to the enclosed gardens of the Middle Ages, and their structure also called to mind the gardens of Ancient Rome. Axial symmetry was reintroduced, as inspired by classical texts.
Sources:
http://gardenofeaden.blogspot.dk/2011/05/rome-opening-times-for-villa-deste.html (Villa d'Este - Rometta)
http://www.louisajones.fr/blog/versailles-and-villa-deste (Villa d'Este - Axis)


