There are different explantion for this background of this Holy Number:
In Buddhism, according to Bhante Gunaratana this number is reached by multiplying the senses smell, touch, taste, hearing, sight, and consciousness by whether they are painful, pleasant or neutral, and then again by whether these are internally generated or externally occurring, and yet again by past, present and future, finally we get 108 feelings. 6 × 3 × 2 × 3 = 108.
More explanations are:
Tibetan Buddhist malas or rosaries are usually 108 beads, reflecting the words of the Buddha called in Tibetan the Kangyur in 108 volumes.
The Lankavatara Sutra has a section where the Bodhisattva Mahamati asks Buddha 108 questions
There are other explanations as well:
The best-known planetary or cosmic derivation is:
There are 12 signs of the zodiac and, in the Vedic tradition, the 9 celestial bodies [Navagraha] as well:
Sun, Moon, Mars, Mercury, Jupiter, Venus, Saturn, plus Rahu and Ketu as the lunar nodes
Multiplying 12 by 9 yields the number 108