We need the meshnet because the Internet isn't good enough.
Ostensibly, it is a network of equals, "peers" who interconnect and exchange traffic freely. Unfortunately, these equals are Tier 1 telecommunications companies, like AT&T or Level 3 Communications. They are at the top of the Internet hierarchy, and the people who the Internet supposedly exists to connect are at the bottom.
The meshnet is a network of people. It is actually net shaped, with each person's node connecting directly to a few other peers, rather than to something "upstream". The meshnet is built and operated by the people it exists to connect.
If you're reading this over your home network, go have a look at your router--the box that says "Netgear" or something on it, with the blinky lights and the antenna. If you look on the back, it has a few network ports. One of them, the "uplink" port, has a cable that leads up the hierarchy. It goes to whoever you pay for "the Internet", a thing which, Netgear seems to think, doesn't include you. The other ports, and the WiFi antenna, connect down the hierarchy to your individual devices. None of the ports lead sideways. If you want to connect your network to your neighbor's, where would you plug in the cable?
Meshnet nodes can connect to each other over any available means of communication: physical cables, WiFi equipment, or even through the Internet. Nodes automatically determine how to route traffic from point A to point B over any and all links available to them. Any two nodes linked together form a fully functional meshnet, regardless of whether they are connected to the broader whole. All meshnet traffic is encrypted in transit; only the sending and receiving nodes can read it.