Hi Subah,
Yeah, I've come across this problem in the past but can't quite remember how I solved it. I think I just used the REPLACE function to either strip out all the apostrophes or replace them with two apostrophes which is SQL Server's escape mechanism for apostrophes. Can't remember which I did to be honest - but it worked.
Another option is to not build the SQL statement dynamically and instead use parameters (standard Execute SQL Task functionality) to insert your ErrorDescription ito your logging table.
Hope that helps.
-Jamie