Why do we apply the holy ash

The ash of any burnt object is not regarded as holy ash. Bhasma (the holy ash) is the ash from the homa (sacrificial fire) where special wood along with ghee and other herbs is offered as worship of the Lord. Or the deity is worshipped by pouring ash as abhisheka and is then distributed as bhasma.

Bhasma is generally applied on the forehead. Some apply it on certain parts of the body like the upper arms, chest etc. Some ascetics rub it all over the body. Many consume a pinch of it each time they receive it. 

The word bhasma means, "that by which our sins are destroyed and the Lord is remembered.” Bha implied bhartsanam ("to destroy") and sma implies smaranam ("to remember"). The application of bhasma therefore signifies destruction of the evil and remembrance of the divine. Bhasma is called vibhuti (which means "glory") as it gives glory to one who applies it and raksha (which means a source of protection) as it protects the wearer from ill health and evil, by purifying him or her.

Tryambakam yajaamahe

Sugandhim pushtivardhanam

Urvaa rukamiva bhandhanaan

Mrytyor muksheeyamaa amrutaat
"We worship the three-eyed Lord Shiva who nourishes and spread fragrance in our lives. May He free us from the shackles of sorrow, change and death – effortlessly, like the fall of a rip brinjal from its stem."