Your computer keeps its IP address. Hamachi just gives you another network interface with another IP address.
Your computer is using it's "regular" Internet connection and IP address (79.x.x.x) to connect to the Hamachi network.
Hamachi has allocated the 5.x.x.x for your VPN connection. This is just another interface.
Your machine never loses it's normal IP address, and that is the address for your non-vpn traffic. So that is what you are seeing when you go to, say, whatismyip.com or whatever other resource identifies your IP address.
If you open Terminal and type
ifconfig
you will see a listing of all the network interfaces for your machine.
When I do this on my Mac, in addition to the "normal" network interfaces (en0, en1, fw0, etc) I also have a ham0, which is the network interface provided by Hamachi. This output also lists my Hamachi addresses.
So, Hamachi just gives you an additional network interface, it doesn't replace the regular interface; which is the one you use to connect to the Hamachi network.
Once you have created and/or joined a Hamachi network, you can "see" the other clients on your Hamachi network via their 5.x.x.x addresses, and they can see you.
I hope this helps.