How to get last truncated time for each tables in SQL Server database
Remember that truncate is an efficiently logged operation.

Since you are running in Simple Recovery mode and provided your T-log is not truncated, you can get it using undocumented fn_dblog:

SELECT [Current LSN]
    ,[Operation]
    ,[Begin Time]
    ,[Transaction ID]
    ,[Transaction Name]
    ,[Transaction SID]
    ,SUSER_SNAME([Transaction SID]) as UserWhoIssuedTruncate
FROM fn_dblog(NULL, NULL)
WHERE Operation = 'LOP_BEGIN_XACT' and
    [Transaction ID] IN (
        SELECT [Transaction ID]
        FROM fn_dblog(NULL, NULL)
        WHERE [Transaction Name] = 'TRUNCATE TABLE'
        )
GO


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Alternatively, you can run a server side trace or turn on SQL Audit.

Always give minimum permissions (principle of least privilege) to your users.


  Is it possible if I change it to FULL ?


Yes it is (and recovery model should be defined by your business - do you want to have point in time restore? How much data your business can afford  to lose? etc.) You have to start maintaining your T-log by running regular log backups. This will keep your T-log at a sensible size.

Edited
Query uses "IN" instead of "=" as multiple results in the sub-query are possible.