Why do most trading platforms require ID Verification?
In the US, these are called "Know your Customer", and "Anti Money Laundering" (KYC/AML) laws and is enforced by FINCEN (financial crimes enforcement network) in the US for any businesses that qualifies as a "Money Service Business" or "Money Transmitter".   

This guidance published on March 2013, http://fincen.gov/statutes_regs/guidance/html/FIN-2013-G001.html , makes it clear that Bitcoin exchanges must comply with existing regulation for KYC/AML.