Why is a common language an important part of nationalism?
Nationalism, at its core, wants everyone to be the same. Having a common language is one big part of expressing that "sameness".
When it comes to how a country views and defines itself, there are different ways the people of that country can view it. Nationalism is one way it can be viewed.
The idea behind Nationalism is that the people view themselves as unified as a single group - as a single type of person. They desire to share the same culture, the same language, the same values as everyone else within the country. It's a philosophy that desires to take away the differences between people, with a typical Nationalist thinking "If I think this way, everyone else thinks this way too. If I act this way, everyone else acts this way too. My values are everyone's values." Nationalists see the country as weaker if there are differences.
It's the opposite of "Melting Pot" philosophy (I'm sure there's a more technical term somewhere, but it's how I think of it), in which the people of the country see themselves as stronger for having differences.
For a Nationalist, a common language is vital, because having people speak the same language indicates there is no difference between people - said in a different way, someone who speaks a foreign language is, by definition, a "foreigner", therefore different, therefore not to be trusted.