Multilayer PCBs consist of three types of layers: there's the core on which the copper is attached, the so-called prepreg layers which insulate the different copper layers from each other, and finally the copper itself. The number of wires depends on the design's complexity; how many connections you have to make and how much room you have for that. Some layers won't be used for routing signals from one point to another, however. On a four layer board the two inner layers are often used only for ground and power supply. This makes ground more reliable, has a shielding effect and decreases the routing from different components to it; ground will often be the net with the most connections.
Those connections between layers are made by vias, drilled metalized holes, so that there's a conductive path between the layers.
Most used are through hole vias, even if you only have to connect the top layer with the second one. That's because they're much cheaper to make: you can finish the PCB and drill through all of it at the end. Buried and blind vias require drilling between the different process steps.
If you have a good look you'll see that the components at the top right are not connected to the surrounding ground.
Circuit design is done at the schematic level, which shows in a logical way how connections between components are made. The PCB is a translation of that conceptual level to the real world. Instead of a rectangle representing a microcontroller you have a specific package with pins. On a schematic you're completely free to place parts in a logical way. On the PCB there are constraints. Pin 23 of the microcontroller is next to pin 24, there's nothing you can do about that. That makes that routing a PCB (connecting everything) can be a difficult task, and there are companies whose engineers are specialized in just that.