Throughout the early to mid 19th Century, Japanese intellectuals and officials watched as China descended into internal turmoil and was increasingly subject to the whims of European powers. Some in Japan began to question the traditional Confucian based ideas on governance, economy, science, etc that Japan had long ago adopted from China. If the once mighty country had fallen so far, maybe their methods were flawed?

In the 1850s, America opened trade between Japan and the west in addition to placing humiliations concessions on the country such as having Westerners in Japan not being subject to native law. This was accomplished with the threat, and occasional use, of force. Of course this intensified the anxiety and discontent of many in Japan.

At this time Japan was still governed by the Tokugawa Shogunate which had ruled since roughly 1600. The shogunate made some efforts to modernize, but grew unpopular and was seen as being ineffectual in the face of mounting western encroachment. A school of thought which said the emperor had to be restored and western methods adopted grew in opposition to the Shogun. Sakuma Shozan, one of the prominent members of this philosophy espoused the slogan "eastern ethics as base, Western technology as means." Such groups alongside broader pro-Emperor and nationalist forces grew violent in their fight against the Shogun. By 1868, after a short war, the Shogun was deposed and the Emperor restored as supreme authority, at least in theory.

The group around the new Emperor Meiji took as part of their mission the idea that "knowledge shall be sought throughout the world so as to invigorate the foundations of imperial rule." This they did with fervor, embracing western methods of industry, finance, military organization, shipping, and to a more limited extent constitutional governance. This was mainly done as the earlier quote suggests, in order to provide a solid and robust state to enforce internal order and outside strength. So although there were no doubt supporters of progress, enlightenment, and advancement as being inherently good, what provided the main impetus for "westernization" was pragmatism. The Meiji leaders essentially didn't want Japan to end up like China during the same period, wracked by domestic strife and reduced to quasi-protectorate status.

Sources:

A Concise History of Japan by Brett L. Walker

The Making of Modern Japan by Marius B. Jansen

Edit

As for the second part of your question, do you mean what made them interested in helping Japan westernize or what made them interested in "opening" the country to trade?