Read that explanation again. The battery voltage must be monitored and when it gets to 4.2 volts, you switch to a constant voltage of 4.2 volts. The battery can never get above 4.2 volts this way. The charging circuit must be two stage. First stage is constant current, Second stage must be a constant voltage of 4.2 volts. It doesn't matter what the voltage might be at the input of the constant current source, that will vary. What matters is the voltage measured across the battery. It must change from constant current to constant voltage of 4.2 volts when the voltage across the battery reaches 4.2 volts during the constant current source. When you first apply the constant current to a discharged battery, the voltage across the battery will measure well below 4.2 volts. As the battery takes a charge, the voltage across it will increase until it reaches 4.2 volts, at which point you stop using constant current and switch to constant voltage.
You cannot properly charge a li-ion with only a constant current source. It must be a two stage charger: First a constant current source and then a constant voltage source.