I'm answering this out of order, and the answer isn't really going to be what you're looking for, but unfortunately it is what it is.
Why do languages undergo dramatic transformations?
There are a number of things that cause languages to undergo rapid change. They are, in no particular order:
Migrations — This is broad. It could be migrations of people from one place to another where the same language is spoken. This could be migrations of other non-that-language-speaking people into the place where the language is spoken. Group A moves to where Group B lives and learns their language. In large enough numbers, it has a big impact.
Conquest — In one regard this is actually the same as migration, in the sense that you have a group of people who become second-language speakers of the language that we're talking about. However there's the added influence of policy regarding what language you can use. Group A moves to Group B's place and tells them they have to speak Language A. Group B are now second-language-learners for a generation and there will be what we call a substratum of Language B in future forms of Language A.
Cultural borrowing — Throughout India there are languages that have borrowed words from Hindi because Hindi-speaking culture is the dominant culture. Hindi has also borrowed a lot of words from English, and earlier on, from Arabic because previously Arabic-speaking culture had an influence. This is not necessarily due to policy, such as in the conquest scenario above, but can often happen just because the other language is popular or trendy. No one is telling the people of Japan that they have to call ice cream "ice cream".
Other rapid lifestyle shifts — Of course conquest causes this for many, but it can happen for a variety of other reasons. Rapid industrialisation, rapid economic development (e.g. South Korea in the latter part of the 20th century), political upheaval that results in a rejection of an older traditional culture (e.g. China in the early 1900s). Any major change will leave a mark on the language.
Other specific reasons exist but those are some of the bigger ones.
One of the major causes of language change is "errors" in how listeners hear what's being said by either the older generation or, in the case of some of the above examples, outsiders who are adopting the language. It can lead to what looks like simplification of a language (but it's not actually simplifying overall in the end; languages don't do that).
What languages have undergone major transformations in the last century?
In fact you would be hard pressed to come up with a language that hasn't undergone major transformations, especially if the timeframe you're suggesting is a century. 100 years is a long time for spoken language. You may point to tiny isolated languages and say that they haven't changed, but this would be incorrect. Even the languages spoken in remote areas, in villages deep in the mountains, have undergone changes.
Now, lacking any major instances of the above points, they may not have changed much relative to some more well known languages, but the changes are still happening. I work with remote minority languages and the communities often talk of trying to go back to speaking "real [whatever]", as they too are just as aware of how much things are changing. Every language is, and it's somewhere between impossible and pointless to try to quantify these changes to determine some relative ranking.
Any claims to the title of "Oldest Language" are bogus. Any claims of "we speak most like how the original language was spoken" are equally bogus. Any claims that "we speak pure [whatever]" are also bogus.
Language change is absolutely a constant, no matter how hard people try to fight it, and unless you're looking at a specific feature of language (vocabulary, for example), it's not something you can really point to and say "This language has changed the most".