What exactly is shown in the 3D cube?

    The easiest way to understand the 3D view is to switch off the PCA axes. In this view, each of the axes of the cube corresponds to one of your samples, and the interpretation is easy: any profile that is high (up-regulated) at the time point corresponding to an axis, is represented by a point that is high (far from the cube's origin) along that axis. You can select which axis represents which sample with the three selectors just below the zoom slider.
    While the interpretation of the image in the non-PCA axes is straightforward, those axes do have a drawback: there is often a steep loss of information in displaying the data. That is because the cube can only show information contained in three samples, while there are often many more than that. As a result, many details regarding the mutual relationships (e.g. distances etc.) among the profiles are lost in that view.
    That is where the PCA comes in: in order to show as much information as possible, the PCA axes "condense" the information in your data, so that most of that information is contained in the three dimensions available to the 3D cube. This is achieved by "rotating" the data in the appropriate way (see the explanation of PCA).

Why do the points in the 3D view seem to form a sphere?

    You are probably using the Pearson Coefficient or the standard correlation to cluster your data, and viewing them in the PCA axes. Since both Pearson and correlation project all data onto the unit sphere, it is obvious that one of the most salient features of the transformed data is that all points are indeed on a sphere. Of course, that sphere has as many dimensions as there are samples, but in the PCA axes you typically see most the information. Accordingly, you also see that your points are projected onto a sphere under Pearson or correlation.