The route from Portugal to Asia was important because of economics. It was important because it represented a way for Portugal to trade for spices directly rather than having to buy them through middlemen. If Portugal could find such a route, its economy would benefit immensely.
In Europe in the 1400s, spices were extremely valuable. They were relatively light and easy to transport and were worth a great deal of money because people wanted them so badly. At that time, spices came to Europe through the Middle East. Italian traders bought them from Muslims in the Middle East and sold them to the rest of Europe. This helped make them rich.
If Portugal could find a route to Asia, it would be able to break the Muslim-Italian hold on the spice trade. If Portuguese sailors could sail around Africa and make it to Asia on their own they could buy spices and sail back to Portugal. That way, they (and not the Italian and Arab middlemen) would reap the profits from the spice trade. This is the main reason why the route from Portugal to Asia was important and why the Portuguese tried so hard to open it.