Yep, VPNs are basically for accessing trusted systems (your LAN) over a public, untrusted network (the internet). Basically, the idea is simply to reuse public wires which have already been laid for the internet, instead of paying for separate, private communication wires (dedicated lines, which cost a lot of money).
The problem, of course, is that, on the internet, you can't trust the wires. Anyone could be listening in on what is sent from your laptop at home to your desktop at work, if it's being sent over the internet.
So, what you do, very simply, is make a "fake" wire (an encrypted virtual network) on top of the existing, public, real wires. You set this up so your computer routes network traffic over it, just like it was a normal network connection.
And that's about it really. The rest is just technical details of how the encryption works, how the routing is done, etc.
p.s.: Wherever I said "wires" (meaning real wires) you can also substitute "radio links" if you have wireless networks.