I feel a bit one-track here, but I think you can again place this on the nefesh habehamis/sichlis line. Children are, by nature, irrational creatures, driven by their emotions. I.e. they are dominated by behamis. In it's 'normal' unpurified state, this is referred to as the yetzer hara that is a part of us (as opposed to the external yetzer hara that even odom had before the chet). 13 is the age when sichlis - which by its very nature pulls us to good, and is therefore the yetzer hatov of a person - begins to manifest, hence the link with starting to keep mitzvos - which relate to nefesh hasichlis - and the spiritual.
BTW here is my translation of a lovely piece that I found in the lexicon of Emunos Vedeos, under the heading 'Sechel Iyuni and Sechel Maasi (p19 in my Israel 1987 edition - can send a pdf if you want the original)
Divided into four levels. The first is the ability to grasp basic priniciples, those which a person knows instinctively, and needs no proof, for example that a whole apple is greater than a half apple, or a diagonal of a square is greater than its side etc. The second level, is the capacity for a person to produce results from general principles, as in grammar (where one produces speech from the grammatical rules) etc. This is called acquired intellect, as it is acquired by a person from first principles and from the senses. And the third level, is the power by which a person percieves the natural form (zurah tivi'is) of all that exists in the lower world, and he will divest it, by virtue of this sechel, from the physical form that bears it, and with this sechel he will meditate on the essence of the soul of man, as long as it is in the envelope of the body. And with (this sechel) he will understand the principles of things and their spirituality. And the fourth power is that with which a person will grasp the forms that are purely spiritual, like the angels, and the essence of the nefesh of man, after its separation from the body, and similar things; all are the fourthe level of sechel iyuni. And all these four levels are called in general sechel iyuni, where the first type of sechel (which deals with speech, and everyday activities) is called sechel maasi, and where this everyday mind is called 'speaker' (medaber), the sechel iyuni is called 'knower'. And in all aspects of this sechel iyuni, it will say only true and false, and it will not say good or bad, for it is not possible to say that the diagonal of a square is larger than the side is a good thing, rather one would say that this is a true thing, and similarly in all its areas....