The MIT license is a very permissive license, and allows re-licensing. The license doesn't require you at all to to keep the attribution line in any of the files, but it's kind of a dick move to remove it straight away. Keeping it there also makes it easier for re-users to find out who the copyright belongs to - though for a license as permissive as the MIT license, which is pretty much compatible with anything, this shouldn't be much of a problem.
In accordance with Should I include a copyright line in every file? it's a good idea to have a header with what license applies and who is the copyright owner in every file.
So who is the copyright owner? You own the copyright to all changes you make, and all the new code you write. For a file you've changed, you have created a derivative work, and each holds the copyright to their own part.
A copy of the MIT license should remain until the work is no longer a derivative. It is not very clear when that is, but at least when none of your code is "based upon" any of the code that was there. If it were to come to it, it may be very difficult to ever prove that it's not based upon any of the old code anymore.
Since your question is tagged license-change, I'm going to assume you want to change not only the attribution, but also the license. With the MIT license, you are free to re-license your derivative work under a different license of your choosing. A good way to do that is to add a second license text file, and refer to that license in a header in each new file you wrote.
In each file that is derived from the template code, link to that license, as well as to the original project and the MIT license.
Copyright (c) 2015 Yeo
Copyright (c) 2014 Konstantin Tarkus (@koistya), KriaSoft LLC
This software is licensed under <whatever license you fancy>.
<Additional header text you may want to include, conventions differ per license>
This software was based on <link to original project>,
licensed under <link to your local MIT license>
When doing that, it is clear what exactly is licensed by whom to whom, and people know what they are getting.