The five classes of stops are ordered by where in the mouth the closure is made, and the sequence runs from the furthest back to the furthest forward: the soft palate, the hard palate, the roof of the mouth with the tongue curled back, the teeth, and the lips. Reciting the twenty-five letters in order is therefore a movement through the mouth in one direction, and a learner rehearsing the sequence is rehearsing the anatomy.
The scheme carries several claims that were not obvious and turned out to be right. That speech sounds can be classified by place of closure. That the same set of features, voicing and aspiration and nasality, applies at every place. That vowels and consonants can be described in the same terms, which is why the vowels are grouped with the classes they share an articulation with. Modern phonetics arrived at all of these independently, much later, and describes them with a similar grid.
What the Indian tradition did not have was a way of writing the analysis except as a list of letters, so the table is both the description and the alphabet. That fusion is the reason the order travelled: a script can be adopted for practical reasons and carry a theory with it, which is more or less what happened when the Siddham letters reached China and Japan.