The answer is that developments in ship and navigational (and, to some degree, military) technology allowed European countries to embark on an age of exploration.
The major change in ship technology was the development of a caravel that was larger than previous ships had been. This larger ship (and its sail technology) allowed voyagers to carry larger cargoes even as they had to bring smaller crews (because of the better sail configurations). This made exploration more economical. This change became even more pronounced later, when much larger ships called galleons were developed.
There were also changes in navigational technology. Europeans started to use quadrants and astrolabes, along with better tables of numbers to use in their calculations, that allowed them to do a better job of figuring out their latitude. They started to use compasses more as well. Finally, they developed ways of making better, more accurate sets of charts and sailing instructions. All of these developments made it easier for them to know where they were and to pass along knowledge of how to get there.
Finally, there was the development of new and better types of cannons. The development of larger ships had allowed for more cannons to be carried on each ship. The development of better cannons made it easier for Europeans to explore far afield with less fear of having their ships taken by pirates or other enemies. All of these technological advances helped make it possible for Europeans to go exploring.