A chord consisting of three notes is called a triad. Triads can be constructed from every scale using the same formula. The triads constructed from a scale are called primary triads and in the root position. The notes in a triad are numbered according to their position in the scale, not the triad. Thus, in the triad ACE the notes are A=1 C=3 and E=5. The triad CAE uses the same notes, so the numbering is still C=3, A=1, and E=5.
A triad progression is two or more arbitary triads played in some sequence. Each triad typically follows the previous within the same octave, although the progression can ultimately move up or down several octaves.
Notes actually name a class of pitch intervals an octave apart. So the A note actually represents all pitches from octave zero, or A0, A1, A2, ..., to An, where n is any integer, usually eight.
In a triad, the notes are ordered in ascending pitch, the lowest called the root note, but without specifying their octave. It is possible to pick an octave for each note and that would still be a triad. For example, the triad ACE could be A2 C5 E7. This arbitary assignment is not precise enough, so we need to provide methods for allocating octaves to the notes.
Put simply, the article is explaining how to assign a specific octave to a note with respect to other note octaves in the triad. Or, how to reorder notes in a triad and what to call them.
Okay so the root typically means the lowest note in a triad. However, the article explains these notes are actually moved up an octave (A2 to A3 for A Harmonic Minor), so they're at a higher pitch than the other two notes in the triad. They are still called the root note (A), even though they are not in the root position (A2). The triad is inverted because the root defines the triad orientation and is now above the other notes (A3) and not below them (A2) as a root normally is.
Often, moving from one chord to the next raises or lowers the overall pitch too much (or not enough). To change that, the notes in either triad are reordered so the new root notes are a (semi-)tone apart. Instead of playing the i and V chords in root position (ACE and EG#B) the V can be reordered to G#BE so the root note change is now A to G#, rather than the jump A to E. Also, both triads have an upper E, which can be suspended (held) between the triads, giving a smoother transition down from AC to G#B.
Similarly, the i chord can be reordered to EAC, so now the i and V share the same root, and the chord change actually lowers the pitch from AC to G#B, while maintaining the same root.
The main idea is to keep control on pitch during chord progression and also develop a flow from one note in a chord to the next. The actual position (relative pitch) of each note in the chord is given an inversion number. The first inversion is the root position with the root above the other two notes.
If this makes sense, you should be able to follow the article without difficulty.