There are a number of factors that contribute to the revival of minority languages.
The most important of these factors is nationalism. Even as globalization occurs rapidly (or in some ways because of globalization) minority peoples do not wish to lose their culture. As world cultures become more similar to one another, minority groups feel it is important that they not lose their culture. This is a form of nationalism.
A second factor is growing wealth. As various groups like the Welsh (or various Native American tribes) come to have more wealth, they have the wherewithal to educate their children in their ancestral language. This was not previously possible when children were not able to get much education.
Finally, there is a growing interest in preserving minority cultures even among the majority. Much of the work of ensuring that minority languages come back has been done at least in part by academics from the majority culture. It is often supported by majority governments.
All in all, then, the revival of minority languages comes about because of changing attitudes and growing wealth.
The first response includes factors that are of great importance, but I would like to add a few more and comment on nationalism, too.
I think that one reason for the effort to save these almost lost languages is, to some degree, the result of the same sort of understanding we are coming to about endangered species. The analogy is not precise, but I think it is worthy of mention. We now know that when a species of fauna or flora is lost, an entire ecosystem is disrupted and harmed, something that is bad not only for the plant and animal kingdom, but for us, too, since we are part of the ecosystem, even though we do not like to think of ourselves that way. There are unintended consequences that are never good. With that understanding, the same sort of thinking is being applied to languages. Each is a part of the history and system of languages, and when we lose one, there are bound to be consequences that affect us somehow, even if we do not know what they all will be. So, it is in the same spirit that we now want to save language.
We do know something about some of the costs of not saving a language, from linguistic, cultural, and historical perspectives. A language that is saved can be studied by linguists, allowing greater insight into the development of languages historically and geographically, a kind of linguistic archeology. When a language disappears, linguists are missing pieces of the puzzle, and this a loss of knowledge to us all. Similarly, culture is embodied in language, as the above response suggests, and when a language is lost, we lose the words that express that culture, much as there are words in English that will disappear because the things they describe have disappeared. When we do not have the language for something, not only has the artifact disappeared, but also the memory of it will disappear. And then, of course, a group has no means, really, of remembering its history. The history of a group is contained in its words. When we lose a language, we lose the knowledge of its relationships to other languages, we lose the unique perspective of a particular culture, and we lose entire swaths of history.
Another reason for the taking back of language is power. One of the first acts of a conquering nation, historically, was to impose its own language on the conquered. So, saving or restoring the language allows some power to be regained, as with the Celtic languages in the British Isles. Separatist movements often go hand in hand with a taking back of language, as in Quebec, in Catalan Spain, or for the Basques.
As far as nationalism is concerned, I think that globalization has made us more inclined to retreat to even smaller units, more tribalism, really, than nationalism. I am not sure we can ever truly go back to being real tribes, but if that is going to be the trend, we will see more different languages, not fewer, a sort of update of the Tower of Babel.