First of all, none of the steps you described are necessary to crack a wireless network. It is the way of working with the Kali Linux VM that the author in the tutorial has chosen. You could also open a terminal in Kali to execute the commands he is running via Putty. If he is using Putty for accessing other Linux machines in his daily work, it is probably the usual way of working with a Linux machine for him.
As you described, he does access the Kali Linux via SSH with Putty in his setup. There are several ways to access a network port of a VirtualBox virtual machine on your host machine.
The way he is using is NAT port forwarding. It is pretty easy to configure if you have already set up a NAT networking interface for your VM.
He is using the port 2222 because it is not possible to bind to ports below 1024 from applications that are not run by root on UNIX systems . (If he is using Windows as his the host system, he could also use port 22.) But connecting to localhost on port 22 with Putty cannot work because the SSH server is forwarded to port 2222.
There other ways, too, such as using the host-only networking  or bridged networking feature . Bridging your Kali VM is a thing you may want to avoid, as it exposes the VM to your local network. As a Kali VM has no firewall configured in the default configuration, this could be a security issue. Configuring a host-only network has no disadvantage in comparison to the port forwarding. A reason he may not do it is the higher complexity in the configuration: you need an additional adapter in your VM and on the host, and you need to find the IP address of the guest after booting.
In the updated rc.d part, he does configure the SSH server to start on boot and
service ssh start to start it now.