For Africa it was the lack of largely developed civilizations that limited them fro any serious explorations. They couldn't explore very far if they didn't have fleets and abundant commissions to fund it.
As for Asia- They did have exploration on a global scale. Gan Ying of Han China in the 1st century travelled all the way to Syria. A distance of about 1/5 of the globe. Zheng He of Ming China in the early 15th century explored all over the Indian sea, even to Kenya where he acquired giraffes to be displayed in the menagerie back in China. A slight few even think that the Chinese discovered the Americas before Columbus, but this is very tenuous in its likelihood. As to why the Asians never definitively made it to the Western hemisphere in their explorations is because the Pacific Ocean is much larger and more difficult to cross than the Atlantic was for the 15-16th century European explorers.