When Vagrant sets up a virtual machine it will set up a port on your local machine that you can use to connect to the box via SSH. By default this is usually port 2222, which Vagrant maps to port 22 on the virtual machine. All this is essentially transparent so that when you type vagrant ssh you connect to the box without any problems. Vagrant handles all of the port matching and key finding behind the scenes.
Vagrant is a tool that allows the easy creation of virtual machines. It was originally developed for use with VirtualBox, but it has been extended to allow integration with other virtualisation tools. Using Vagrant you can create a particular setup that you can then share with other people without having to give them large virtual disk images.
I have to admit, when I first heard of Vagrant I didn't 'get it' and wasn't sure why I should be using it at all. Since then I have done lots of reading and talking with other developers and I am now wondering how I managed to work without it.