Which form of Shiva is it in this picture? Why did he take this form?
He is Swacchandanatha form of Lord Shiva who made Tantras.
From the Lecture on Practice and Discipline in Kashmir Shaivism 1982 - Swami Lakshman Joo Raina says:
-BIRTH OF TANTRA-
In the beginning of Satyuga, Lord Shiva appeared in the form of Swacchandanatha, with his five mukha(mouth), namely Ishana, Tatpurusha, Sadyojata, Vamadeva and Aghor. These mukhas represent his five energies, namely, Consciousness, Bliss, Will, Knowledge and Action. He manifested Tantras through his five mukhas (energies), some given by each of his mouth while others recited by him through two or more than two mouths simultaneously.
The Tantras that came into existence are Dualistic and Mono-Dualistic and are known as "Shiva Tantra (10) and Rudra Tantra (18)" respectively.
When these five energies of Lord Shiva unite with each other in such a way that each of these take hold of the rest simultaneously, they give shape to 64 Bhairav Tantras which are purely Monistic(Advait). The thought expounded in these Tantras is called the "Trika" philosophy or the Kashmir Shaivism.
This picture represents Pachamukhi/ Panchavaktra Shiva.
According to Pashupata Shaivism, there are five aspects/ forms of the Shiva. These five aspects are represented by five faces of Shiva. Each face represents different direction, element,time,senses of the human body, energy and color.
The first face of the Shiva is called Sadyojata (Revealer). It is a combination of two words'Sadyaha' which means just now and 'Jataha' which means now born. It is facing towards west. It is showed in white color. This aspect is present as mind in all beings. It is representing fire element. Smell is the manifestation of this aspect. This form is existing in our body as sense of smell and organ of reproduction.
The second face of Shiva is known as Vamadeva (Concealer). Vama means "left", which is the seat of the shakti and is associated with the feminine, and deva means "God"). It is represented in red color. This is displayed in half male and half female representing ardhnari. It is directed towards north. Present in all beings as ego/identity. It represents the element water. In our body, this aspect is present as the sense of taste and the organ of excretion.
The third face of Shiva is Aghora or Bhairava (Destroyer). It is black in colour. This face is directed towards south. In all beings, this aspect is present as intellect. The element fire represents Aghora. In our bodies, this form is manifested as sense of vision and our feet.
The fourth face of Lord Shiva is called Tatpurusha (the Cosmic being). It is having golden complexion and is facing towards east. Represented in element of air. This form is present in our bodies as the sense of touch and hands.
The fifth face is called Ishana (the Lord). It is represented by color similar to copper. This aspect represents consciousness. The is manifested by physical element of space. Sound is a manifestation of this form. This face is directing skywards. In our bodies it is present as the sense of hearing and the organ of speech.