I think you want to sum the age of person having [id],[name],[age],[month]. If so then there is nothing wrong with your query. But i guess id can't be duplicate in your table. Please remove ID field and test your query you will defiantly got sum of age.
SELECT * FROM (SELECT [name],[age],[month] FROM [testing].[dbo].[test_1] )AS PT PIVOT(SUM([age]) for [month] in ([March],[April],[May],[June],[July],[August])) as [result]