What is the SPIRITUAL explanation to Shiva's manifestation as Sri Dakshina Murthy?

One of the most popular forms of Shiva worshipped in South India is the sage Dakshinamurthi, whose name literally means "the South-facing form". 

Dakshinamurthi is the subject of the famous Dakshinamurthi Stotram, a hymn commonly attributed to Sri Adi Shankarcharya.

चित्रं वटतरोर्मूले वृद्धाः शिष्या गुरुर्युवा ।गुरोस्तु मौनं व्याख्यानं शिष्यास्तुच्छिन्नसंशयाः ॥३॥

It is indeed a strange picture to behold; At the root (i.e. base) of a Banyan Tree (Vata) are seated old Disciples (i.e. aged Disciples) in front of an Young Guru,

The Guru is Silent, and Silence is His exposition (of the Highest Knowledge); and that (Silence) is severing the doubts (automatically) from the minds of the Disciples.

Dakshinamurty form of Lord Shiva is described in Suta Samhita of Skanda Purana in Mochak Kathanam of Mukti Khanda.

----

Now a question arises as to the significance of Shiva's manifestation as Sri Dakshina Murthy.

---

Shiva means "the auspicious/kind one". The word Shiva is used as an epithet for several Rigvedic deities, including Rudra.

Like other epithets like Indra, Rudra, etc, Shiva is an epithet used to indicate the auspiciousness/kindness of the Almighty God.

For example:

In respect of Agni: (RV I.31.1)

तवमग्ने परथमो अङगिरा रषिर्देवो देवानामभवः शिवः सखा | तव वरते कवयो विद्मनापसो.अजायन्त मरुतो भराजद्र्ष्टयः ||

You, Agni, the frst Aṅgiras, the seer, the god, became a kindly comrade of the gods. Under your commandment were born the sage poets working with their know-how—the Maruts with glinting spears.

In respect of Indra : (RV II.20.3)

स नो युवेन्द्रो जोहूत्रः सखा शिवो नरामस्तु पाता | यः शंसन्तं यः शशमानमूती पचन्तं च सतुवन्तंच परणेषत ||

May Indra, called with solemn invocations. the young, the Friend, be men's auspicious keeper, One who will further with his aid the singer, the toiler, praiser, dresser of oblations.

---

Coming to the aspect of Shiva's manifestation as Sri Dakshina Murty, as a Guru of all Sages which are like embodiment of Dharma tree and who have transcended even Vedas, they are sitting around that Banayan tree, in my opinion, this concept is referring to manifestation of Almighty God.

We have to remember that a Guru of all Sages, which are like embodiment of Dharma tree and who have transcended even Vedas, can only be the Almighty God.

It does not indicate manifestation of Sankara or Husband of Uma as Sri Dakshina Murty, as the term Shiva is an epithet of the formless Almighty God.

---

Further, Dakshina Murty doesn't indicate facing Southern direction, but indicates propounder/follower of Vedic way of life.