You want insert into . . . select:
INSERT INTO [DB_A].[dbo.a_test](a,b,c,d,e) --ADDED A COLUMN
select p.product_info, p.product_date, p.smth, pr.program_name, pr.program_smth
FROM [DB_B].dbo.products p LEFT JOIN
[DB_B].dbo.program pr
ON p.program_name = pr.product_info;
I also fixed the query to use table aliases, so it is much easier to read.