The emitter current (which is your LED current) is the sum of the base and collector currents. With the collector disconnected, you effectively have a voltage+resistance+diode+LED loop, resulting in a base current of about Ib =(5-0.5-1.5)/10680 = 0.3mA, which results in a dimly glowing LED.
Because of that 0.3mA base current, the transistor will be pretty strongly "on", meaning the collector will conduct (a lot more) current than the base. If you then connect a voltage source between the collector and the bottom of your existing 5V source, then additional current will flow through the collector and the LED.
For the chosen transistor in this configuration and a sane collector voltage (say 12V), the transistor will probably be saturated, which means the Vce (collector-emitter) voltage will be at a minimum, and the current flowing through it will be determined primarily by the rest of the circuit. In this case, say you have Vc = 12V, that gives Ic = (12-0.7-1.5)/680 = 14.4mA.
Now the emitter current Ie = Ic + Ib = 14.7mA and you have a nicely-glowing LED.
where the random-looking numbers come from:
Vbe = 0.5V (transistor base/emitter acting as diode)
Vcesat = 0.7V (saturated collector/emitter voltage)
Vf = 1.5V (LED forward voltage)