What exactly is the purpose of this yantra?
First, we need to know what the purpose of a Yantra in general is.
In Hinduism, although the God is essentially stated to be Nirguna, but Saguna worship is mainly recommended, as worshipping the form-less is basically an impossible task.
And, for that purpose, symbols (Pratik) are introduced in various scriptures, in which we invoke the presence first and then worship the deities.
In Vedas, fire is the mode, in Upanishads Om is the Pratik, in Puranas it is idols (PratimA) and in the Tantras it is (primarily) Yantras.
Yantram mantramayam proktam devatA mantrarupini |
Yantre sA pujitA
devi sahsaiva prasidAti ||
O Devi, know the Yantra to be Mantra-mayam (mantra-endowed) and the
Deity to be of the form of mantra. If worshipped in Yantra, the deity
is appeased instantly.
KulArnava Tantram 6.86
Besides that, Tantras also say that unless worshipped in a Yantra, the deity is not appeased.
Yet another verse, which indicates the inherent identity between a deity and it's Yantra, is as follows:
Shariramiva jivasya dipasya snehavat priye |
tathA yantram pratishtitha ||
O Beloved, just like the Jivas have the bodies, the lamps the oil and
ghee, likewise, for all deities are laid down their respective
KulArnava Tantram 6.88
So, a Yantra is basically a dynamic graph which can be identified as the respective deity's body. Also, as it is Mantra-mayam, it is also endowed with the divine consciousness of the deity.
So, in other words, there is no difference at all between a deity and it's Yantra. Whatever worship that is offered to a Yantra goes to its respective deity.
Now, it should be easy to understand what Goddess Bagalamukhi Yantra represents.
Who was the first person who used or we can say was originator of this
It is Lord Shiva of course, because he has created all the Yantras that we have. Including the various Yantras of his consort.
Brihat TantrasAra gives the following verse guiding how the Yantra is formed:
Tryasram sharasram vrittam astadala padmam ||
First draw a triangle pointing up; over it draw a Sharasram - a
six-pointed star which is formed by drawing two triangles one pointing
up and one pointing down; over it draw a Vritta - a circle and on it
draw an eight-petalled lotus. Finally draw the Bhupura (the squarish
enclosure) over the configuration.
Any other interesting facts (worth sharing) associated with this
The basic purpose of Bagalamukhi worship is to achieve Stambhana, one of the Shat Karmas of Tantras, which basically means to completely paralyze a wicked enemy.
From this answer:
Lord Shiva says the following about the purpose of Goddess Bagalamukhi's mantra and worship:
BrhmAstram SampravkshyAmi Sadya Pratyayakarakam |
HitArthAya StambhanAya Cha VairinAm |
Pavanohapi SthirAyate |
For the benefit of the devotees and for paralyzing the evil groups
of enemy (vairi) I am revealing the Bagalamukhi mantra, which is as
powerful as the Brahmastra. This mantra can paralyze anyone or
anything. Even the movement of air can be stopped by its mere
remembrance, what to say of others.
So, the purpose is Stambhanam or completely paralyzing (or subjugating) one's evil enemy.