The major and minor scales (or, as many commonly associate it with, "Do-Re-Mi-Fa-Sol-La-Ti-Do"), are the foundation of all tonal music, and can be considered the building blocks of almost all forms of music written prior to the 20th century. Understanding how to aurally identify and reproduce each pitch of the major and minor scales can become fundamental in the music student's capacity to understand music theory.
Chord arpeggios are linear reflections of vertical triadic harmonies. This means that, though a "chord" is technically considered a musical occurrence when two or more notes of a scale sound at once, chord arpeggios are when the notes of a chord are played or sung one pitch at a time. The result is a melodic line composed of specific pitches of a scale, in which the quality of the chord can vary depending on which pitches and which scales are being used. Understanding how to aurally perceive this concept will help clarify what may initially seem like an extremely complex idea.
Learning to hear and sing diatonic intervals (or, in other words, the distance between two notes of a scale which do not occur immediately next to one another in the scale) can help the music student understand how melodic contour is created. Since certain intervals are more dissonant than others, the musical function of each interval is unique. Understanding how to hear these intervals can help the music theory student develop a strong aptitude for melodic part writing.
The ability to dictate a melodic line as a tool of ear training can be useful in the context of music theory because it promotes an understanding of what musical theories are used to write a certain melodic line. If the student is familiar with the most commonly used concepts in melodic part writing, then he or she will also be quite capable of hearing and dictating these concepts as well.
The importance of harmonic dictation is very similar to the importance of melodic dictation, except that harmonic dictation promotes an understanding of vertical occurrences (i.e. the traditional concepts used in sounding more than one note at a time). As with melodic dictation, if the student can understand what traditional, vertical harmonic progressions to listen for, it will likewise promote an understanding of the music theory involved in creating these traditional progressions.