Current Sources - Usage and Projects
The page you point to uses a current source to drive an LED which is a very popular
real-world application and is also a good weekend project ;)
The color temperature of the LEDs changes with current so keeping the current 
constant is useful for some applications. 


Current sources can also be used in battery charging applications where you
need to maintain a constant current to properly charge a particular chemistry 
of battery. 
I have used current sinks to test power supplies. A typically power supply test
will be to run the power supply at it's rated current to verify proper voltage
regulation.
Your multimeter may use a current source to measure resistance. Pass a current
through an unknown resistance and measure a voltage.
Within circuits a current source (and sink) can be used to bias transistors (like
a differential amplifier circuit).