Why did the europeans begin to explore overseas?

Firstly, the Portuguese had recently designed the caravel; a small, maneuverable ship developed in the 1400s to explore West Africa. It used lateen sails, which allowed those who sailed to catch the wind when necessary, as well as effectively sail against the wind. Developments in cartography allowed for more accurate navigation. They also had the astrolabe, which was originally critical for navigating. For the first time, thanks to Prince Henry "the Navigator", they did not have to sail in close proximity to the shoreline; they could venture out without too much concern of navigation. They were also funded by the Prince, who encouraged exploration and colonization.

At first, they were attempting to find a route to East Asia, because spices were highly coveted. (Think about it--spicy foods boost your metabolism, make everything taste better, and were only available on one continent.) They had no idea how far down Africa stretched. Eventually Vasco de Gama reached India.

The Spanish had the same idea; they desired spices, gold, silver, and other precious metals that were believed to only be found in Asia.

Additionally, the desire to claim further land and spread Christianity were prominent during the Age of Discovery.

Famously, Christopher Columbus's voyage hit a wind that took him to not round the Cape of Good Hope as intended, but instead to the Latin American coast and to Florida. This sparked new voyages to colonize land.

Specific to the English: after failing to find a quick route to Asia, and the defeat of the Spanish Armada (which took place after years of Spanish expeditions to colonize South America, Latin America, and unsuccessfully, Western present-day America) opened up the opportunity to colonize overseas and encroach on Spain's formerly claimed territories.