That study uses a crossover design, which means that everyone gets all three treatments (in this case, phenylephrine, pseudoephedrine, and placebo), but in different phases. You might get the placebo this week and the pseudoephedrine next week, while I get the opposite. Each one of these time steps is a "phase".

A similar study might use a "batch design" instead, where the subjects are each tested once, after being given a single treatment. They might put you in the pseudoephedrine group, while I get the placebo.

Crossover designs have a big advantage in that are more robust to individual variability, and thus have more statistical power to detect differences. In essence, the crossover design allows you to compare the average of each subjects' difference between conditions while the block designs force you to examine the difference between the average scores of each group.

This power comes at a price--you have to be careful that the different treatments do not interact so you can correctly associate causes and effects. None of the treatments remain in the subjects' system for a week, but they are worried that people might remember* how well pseudoephedrine worked and "downgrade" the other treatments. One way to check this is to analyze the first phase of your data, where each subject has had only one treatment, as a batch design, which is what they reported there.

* The measurement here is a self-reported scale of nasal congestion. Note that they don't do this for most of the quantitative/physiological measurements.