By the late 1400s, technology had improved enough that it was more possible for Europeans to explore far overseas. The technological advances were partly in ship design and partly in navigation.
One thing that helped allow exploration was bigger and better ships. The designs most often mentioned in textbooks are caravels and galleons. Both of these kinds of ships (caravels came first) were improvements on what had come before. They were larger and could carry more cargo and cannon (for protection). They had better sail designs and could sail more efficiently and with fewer crew members. These new kinds of ships made exploring more feasible.
In addition, there were changes in navigational technology. These included quadrants and astrolabes for determining latitude. It also included better tables of data that told navigators what their astrolabe readings meant. It included compasses and better types of charts and sailing directions (called rutters) for telling people where an explorer had been and how to retrace his steps.
These types of technological improvements made it more possible for Europeans to explore far away from their own shores by the late 1400s.