It mostly comes down to productivity and efficiency. It is a huge pain to install an OS in a virtual machine and then manually get the updates installed regularly, for let's say 20 machines.
Also this can be even more difficult when we are testing historic releases.
From the web development and testing stand-point a key benefit is you are assured you are using exactly the same environment in design, as the developers and testers are.
As John Boyd said of the Observe-Orient-Decide-Act (OODA) loop
we need to respond to change more quickly than everyone else. Vagrant allows us to tighten that loop.
So in summary not only does it make life easier when we are working with older releases, from the configuration standpoint. It saves time in deploying the test enviornments, ensures everyone is 'on the same page' designer, developer, and testers and makes it easier for us to respond to that all vital change just fractionally ahead of the curve.
References
MSOpenTech on Vagrant Technology and
Wikipedia - OODA Loop
Edit: In relation to comment
Vagrant makes things easier. You no longer have to use nano to configure your virtual machine or sit there for hours stuck at a command line, wondering if you colleague is using a LAMP stack or LEMP stack.
You save time, and you gain the ability to mirror identical environments to everybody working on the project. You can version control the machine configuration, and roll back easily.
If you're working in larger environments with load balancing etc you can build all your servers simply by adding them to the vagrant file.
Issues such as a colleague having a different version of nginx installed, or php x.x.x instead of x.x.y are a thing of the past. You can check in your changes, have your colleague check them out and know they will be running them on exactly the same server that you have just finished developing on.