We can easily dismiss the mystic's attraction to asceticism, abstinence and renunciation as originating from backward life- and body-denying ideas, that are no longer relevant to our enlightened day and age. To make it more interesting, I suggest we put aside that attitude for a moment, and consider other possibilities.
First, some facts: In many of the religions, East and West, people who choose to dedicate themselves to the spiritual/religious life willfully renounce things that are considered desirable in the secular society. In some religions they renounce the accumulation of personal property, the freedom to do as they like, comfort, good food, family, sex, career, etc. Poverty, reclusion, fasting, celibacy and other forms of renunciation are common among practitioners of different mystical paths.
Now, a personal story. When I was at the end of the fifth year of my medical studies in Jerusalem, I underwent "an event of unitive consciousness" - direct knowing of unity as the ultimate reality. This knowing dissolved at once the main existential questions I had. When I "returned" to a dualistic state of consciousness, it occurred to me that this knowing could be for me a transient experience - or the foundation of a new life. I knew: my response to it would determine the outcome. If I wanted my life to be a reflection of this knowing, I had to give up something of value, as an expression of and testimony to that intention. The greater the sacrifice, the more power I would be giving to my intention. At that moment I decided to leave medical school and let go of my dream of a lucrative career. I followed up on that decision and never looked back.
Back to the general topic: The mystic's attraction to asceticism, abstinence and renunciation I understand as driven by his or her desire to give a concrete, tangible, "external" expression to the spiritual impulse / love of God / longing to merge with the Absolute that they feel. They want to see with their eyes what they feel in their heart. Such actions make that possible. It worked for me.