Best How To :

SQL Server does not implement the xPath function string-join, so you would need to adopt a two step process, the first would be to extract all the terms to rows using nodes();

SELECT  n.value('.', 'VARCHAR(100)') AS parsedString
FROM    #temp AS t
        CROSS APPLY t.Response.nodes('/error/description2') r (n);

Which gives you your values as rows:

parsedString
----------------------------------------------------------------------------
Country Code is required
Error While Saving the Project info
Smart Rule PROJECT_COUNTRYCODE_VAL Violated: Country Code is required
Error While Saving the Project

Then you can add your delimeter and concatenate them back up, using FOR XML PATH(''), TYPE, and finally use STUFF to remove the first delimeter:

SELECT  STUFF(( SELECT  ',' + n.value('.', 'VARCHAR(100)') AS parsedString
                FROM    #temp AS t
                        CROSS APPLY t.Response.nodes('/error/description2') r (n)
                FOR XML PATH(''), TYPE
            ).value('.', 'VARCHAR(MAX)'), 1, 1, '') AS ParsedString;

FULL WORKING EXAMPLE

DECLARE @X XML = '<error>
  <errorno>BL04002055</errorno>
  <description>Smart Rule PROJECT_COUNTRYCODE_VAL Violated</description>
  <description2>Country Code is required</description2>
  <correction />
</error>
<error>
  <errorno>BL01001973</errorno>
  <description />
  <description2>Error While Saving the Project info</description2>
  <correction />
</error>
<error>
  <errorno>Unable to Create Custom Object</errorno>
  <description />
  <description2>Smart Rule PROJECT_COUNTRYCODE_VAL Violated: Country Code is required
Error While Saving the Project info</description2>
  <correction />
</error>';

SELECT  STUFF(( SELECT  ',' + n.value('.', 'VARCHAR(100)') AS parsedString
                FROM    (SELECT @X) AS t (Response)
                        CROSS APPLY t.Response.nodes('/error/description2') r (n)
                FOR XML PATH(''), TYPE
            ).value('.', 'VARCHAR(MAX)'), 1, 1, '') AS ParsedString;