You can use a Union query, for example:
INSERT INTO NewTable (Source, Plate)
SELECT Source, Plate FROM
(SELECT Source, Plate1 As Plate FROM Table
UNION ALL
SELECT Source, Plate2 As Plate FROM Table
UNION ALL
SELECT Source, Plate3 As Plate FROM Table
UNION ALL
SELECT Source, Plate4 As Plate FROM Table) As t
EDIT: the Union part by itself:
SELECT Source, Plate1 As Plate FROM Table
UNION ALL
SELECT Source, Plate2 As Plate FROM Table
UNION ALL
SELECT Source, Plate3 As Plate FROM Table
UNION ALL
SELECT Source, Plate4 As Plate FROM Table
Or
SELECT Source, Plate FROM
(SELECT Source, Plate1 As Plate FROM Table
UNION ALL
SELECT Source, Plate2 As Plate FROM Table
UNION ALL
SELECT Source, Plate3 As Plate FROM Table
UNION ALL
SELECT Source, Plate4 As Plate FROM Table) As s