Why does heritage conservation be so important? 

The significant or importance of expression "Heritage Conservation" is divided from its definition in the Nara Document of Authenticity / Appendix no. 2 that "Conservation: all efforts designed to understand cultural heritage, know its history and meaning, ensure its material safeguard and, as required, its presentation, restoration and enhancement".
Hence, heritage conservation is the main tool to preserve and transmit an intangible heritage behind each heritage context to the future generation. Respectively, without this significant process, the heritage site couldn't act the pattern of sustainability.
Respectively, the legislator asserted throughout UNESCO Operational Guidelines for the implementation of the World Heritage Convention that in paragraph no. 6 "The protection and conservation of the natural and cultural heritage are a significant contribution to sustainable development". Also, in paragraph no. 32, we can notice that one of primary functions of ICCROM (the International Centre for the Study of the Preservation and Restoration of Cultural Property) is "to carry out …… public awareness programs to strengthen conservation of immovable and moveable cultural heritage".
 
Then, the legislator asserted in the Nara Document of Authenticity / paragraph no. 4 that

In a world that is increasingly subject to the forces of globalization and homogenization, and in a world in which the search for cultural identity is sometimes pursued through aggressive nationalism and the suppression of the cultures of minorities, the essential contribution made by the consideration of authenticity in conservation practice is to clarify and illuminate the collective memory of humanity.

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What is the role of the civil society?

Actually, we should divide an expression "Civil Society" to two entities:
Firstly, associations, initiatives, and non-governmental organizations;
Secondly, the main owners of the cultural heritage "Citizens".
Consequently, we can understand the role of the civil society through an interaction between two entities especially throughout the heritage cycle of Simon Thurley.
Furthermore, through UNESCO Operational Guidelines for the implementation of the World Heritage Convention / paragraph no. 119:

"Legislations, policies and strategies affecting World Heritage properties should ensure the protection of the Outstanding Universal Value, support the wider conservation of natural and cultural heritage, and promote and encourage the active participation of the communities and stakeholders concerned with the property as necessary conditions to its sustainable protection, conservation, management and presentation."