A rare instance where ChemGuide uses some clumsy language.
The line in a phase diagram is where the standard Gibbs energies of the two phases are exactly equal, so there is no driving force for one phase to turn into the other (or vice-versa). So let's say you are in the liquid phase at a certain temperature. There is always an equilibrium between the liquid and solid state, and the Gibbs energies of the two phases are temperature dependent, so you can calculate a "rate of evaporation".
For instance, let’s look at the standard Gibbs energy change (J/mol) for water going from liquid to vaporous state (calculated from heat capacities) as a function of temperature with the pressure held constant at one atmosphere.
You see quite clearly that there is a change in sign of the standard Gibbs energy change at about 373 K, or 100 Celcius, right where you know the boiling point is.
Now let’s take some other point, say 353.15 K. There is still an equilibrium going on at this temperature. The vaporization of liquid water is not thermodynamically spontaneous at this temperature, but even so there is still an equilibrium, so we know that there will always be some water present as water vapor. So if it’s not-spontaneous, why does a puddle of water evaporate to completion even at room temperature? Well that’s because of La Chatellier’s principle – the vapor can drift away, which continually drives the equilibrium toward vaporization, even though it’s not a spontaneous process. Still, the standard Gibbs energy change becomes less positive as the temperature increases, which naturally predicts what we know to be the case: at higher temperature, vaporization will happen faster, because the equilibrium more and more favors the gas state.
As we go from 353.15 K and increase the temperature, at around 373.15 the sign of the Gibbs energy change switches sign from positive to negative. Then the process of vaporization becomes spontaneous. All this really means is that the gas state is favored over the liquid state. Practically, nothing has really changed – there is still and equilibrium, so you will still have liquid hanging around. Of course, boiling is a little more complex than this because of cavitation and so on, but don’t make your life too difficult.
Anyway, point is that that there is a temperature where the standard Gibbs energy change is zero, because the liquid and gas states have the same standard Gibbs energy. At this point the gas and liquid are equally favored and there is 1 to 1 equilibrium (K = 1). The line in the phase diagram is made by plotting out where this temperature resides for every pressure.