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Looking here, I am shown that the connecting symbol, means the wires are, literally "connected" as shown below.
I am currently working on trying to decipher an FM Radio schematic, but am having trouble with this "connected" thing...
On the above schematic, I am shown that almost everywhere on the board, there are connecting wires.
Is this exactly how it's meant to be soldered/constructed? Am I meant to grab 3 wires, and just drop a blob of solder on them so that they are all connected?
It might be easier if you understand that schematics are simply a symbolic representation of the circuit, the components do not need to be placed in that particular physical layout to work, i.e. R2 doesn't have to be vertical and R3 doesn't have to be horizontal (though it is often easy to think of it as a close approximation to the layout). As for "connected", it could mean they are soldered together, either through wires, the leads of the components if using through hole, or are both connected to the same trace of a PCB at some point. It could also mean they belong to the same "node" on a breadboard, in which case there is no solder.
So for example with TR2, the collector at some point in following the path (it could be wires, leads of components, traces on PCB, etc.) of the collector, it must be connected to one end of each C4, R3, and R4. The base must be connected to the other end of R3 and one end of C3 and the emitter is connected to ground.
The schematic shows how components should be electrically connected. It does not show how the connections should be made when the circuit is actually constructed (although sometimes special requirements may be pointed out). Construction requirements, that is the layout of the components and the wiring, are dependent on the type of circuit. For low frequency and low power circuits, the wiring can be fairly sloppy because parasitic capacitances and voltage drops in the wiring are not critical. As the frequency and the power levels go up, wiring becomes more critical. Often a circuit description will be accompanied by construction notes. If not, you will have to use judgement and experience to determine how the circuit should be constructed in order to work properly. I suggest looking at actual equipments to see how the layout and wiring are done compared to the frequency and power levels involved.
Connected means electrically connected; that usually means soldered together, but not necessarily so. Applying mechanical pressure on two conductors will make them connected, and that's the principle used for most connectors.
You can simply solder the components together as you mentioned, but you'll find that, except for the simplest circuits, to be extremely difficult; it's hard to keep them in place when the only thing that's mechanically supporting them is also the solder that connects them electrically, not to mention unwanted connections when two wires accidentaly cross. For development/testing/prototyping, use a breadboard. When the circuit is working correctly, transfer it to a PCB.
A "schematic" shows the logical connection between devices, not the physical connection. You can use any construction that provides equivalent connectivity. Usually there will be a separate layout diagram that shows how it should be constructed, such as on a PCB.
Solderless breadboards are popular but have problems at radio frequencies. Radio circuits can be built with the "deadbug on copper clad" technique (http://blog.eepro.to/2013/01/example-circuit-boards.html)
Look at 
The large copper ground plane connects all the components soldered to it together. That would correspond to the bottom line of your circuit diagram. The circuit is made among the components soldered together in mid air, not touching the copper board. An "island" has been cut (marked "output") that is also not connected to the copper ground plane.
QUICK ANSWER "NO"
The others here are good and not wrong, but in short.
Connected DOES NOT mean "solder together", connected means "a path exists between the two", you could use a wire, any conducting material.
For example "my hard disk is connected to my computer" - it's not physically soldered to it but there is a connection
Less far-fetched example "connect the power LED to the motherboard" - no soldering at all, and infact there are a pair of wires between the LED and motherboard.
As you get to higher-frequencies or higher power wires behave less like these "perfect vesseles of current transmission" and start to act as resisters and tiny capacitors, then a wire doesn't "connect" as such because it adds resistance based on it's length, but for your low power low frequency stuff wires are magical current transmission devices that do not distort or absorb or anything.
