Portuguese exploration down the coast of Africa started for earnest in the 1430s, although the symbolic date is usually given as 1415 of the conquest of Ceuta on African shore. So the affair took from 60 to 75 years to reach the southern tip once they started.

With this in mind, your question can be interpreted in two ways, why did it take only until 15th century for Europeans to start this exploration, and once they ventured why did it take them roughly three quarters of the centuries to complete? The two are actually somewhat connected.

Lets start with the first, why was it only the fifteenth century the exploration started. There are some geopolitical and macro-economical factors at work here. Namely, it is the 12th and 13th centuries that saw the rise of Italian merchant republics, as well as Christian Iberian kingdoms conquering the Southern parts of their peninsula. That, with other developments paved way for regular seaborne commerce through the straits of Gibraltar between the Mediterranean and the Atlantic/North sea which was established only towards the close of 13th century. And this regular economic activity facilitated the necessary economic background required for the commercial activity in the oceans. As it happened only by the 14th century it would be understandable why it didn't occur earlier.

There are also some environmental and technological reasons usually given for why the exploration started so late, but I would be very careful how much importance is put into those. For example, it is true that after passing Cape Bojador once can find currents and winds that make it very hard to return back to Europe, and after passing it for the next 1000 km the coast is basically a sparsely populated desert. And certainly the ships of fifteenth century were better suited for this travel and could finish it with more ease and less hardships. But it is not like those currents were insurmountable, and that previous ships couldn't under any possibility do it. They could, it just made it harder which put off many against even trying. The Europeans in general, knew Africa was there but had no idea of its shape and size further south. Any expedition was into unknown, and by that metric: hard, dangerous, risky and - which is quite important - expensive. There were few willing to risk such a thing, but it did occur from time to time. Famous example is Genoan Vivaldi brothers attempting in 1291 (on the wave of the above mentioned successful forays in to the Atlantic) to circumvent Africa and reach India. However it failed, and the failure actually remained in memory for some time.

So we are now reaching the Portuguese and 15th century. I've talked about the various reasons they became interested in Africa but for here it is important to highlight their slow, methodological, incremental long trek down the coast. While we may puzzle why it took them several decades and wonder if they could have done it faster; their method had much benefits. With each voyage going only slightly further down the coast before returning, they were accumulating knowledge and information (by mapping and capturing locals) about the regions making each further trip a bit more safe and less riskier. Also, each voyage wasn't purely exploratory - although ships had instructions to explore as much as they saw fit - but also commercial. Most voyages were successfully earning money - by bringing back slaves, gold, ivory and other goods - some acquired by trade, some by raid and plunder. In each case it resulted in nature of those voyages changing from being a major risk of huge amount of money, to a increasingly reasonable investment with diminishing risks and high returns. It created a sort of whole sustainable "economic" sector that earned money from investing in exploration/trade down Africa, perpetuating itself and bringing in talent and innovations .

Like all investments and economic markets it had its ups and downs, but coupled with direct royal involvement and encouragement from time to time, enough to restart the whole thing if things wound down - you got a successful multi generational success in going around Africa and reaching India.