This is because before downshifting, the RPM in the new gear will be higher than the RPM your engine is currently spinning at, so unless you want a lurch when you let out the clutch, you blip the gas to match up the revs. The amount of throttle needed for the blip depends on the spacing of the gear ratios and current RPM, and executing this technique smoothly takes skill and shows excellence in manual transmission driving ability.
It is not necessary when driving normally as most of your downshifts will involve relatively small RPM changes, i.e. braking, then changing from 4th to 2nd and letting out the clutch at 1200rpm to take the right turn. However, it is used in racing and when driving at higher RPMs, as a downshift from 4000RPM in 4th gear will often place you at 5400RPM in 3rd, and without blipping, the engine speed will be closer to 3500RPM due to revs dropping.
Chris was driving his 2005 STi on the Nurburgring, but did not know how to heel-toe, so every time he braked and downshifted for a corner there was a jerk when he let out the clutch.
Jessica was bored and decided to heel-toe downshift through all the gears just for fun as she approached a red light, giving a blip with every downshift.