You can use ST_GeometryN together with ST_NumGeometries and the generate_series function to obtain what you need.
Let's assume you have the table from Jakub's example:
CREATE TABLE multi AS(
SELECT 1 as id, 2 as test, ST_GeomFromText('MULTIPOLYGON(((0 0,4 0,4 4,0 4,0 0)),((1 1,2 1,2 2,1 2,1 1)), ((-1 -1,-1 -2,-2 -2,-2 -1,-1 -1)))') AS geom
);
This one contains a multipolygon, an id and another column.
To get each single polygon from the table including all other attributes try something like:
SELECT id, test, ST_GeometryN(geom, generate_series(1, ST_NumGeometries(geom))) AS geom
FROM multi
"id" and "test" are the values for each row in the original table.
generate_series creates a series of numbers from 1 to the number of geometries in each row.
Therefore you will split each multi geometry in its separate single geometry parts and the values in the other columns remain the same.
Just replace the columns and table in the example with the columns from your exported shapefile and you will get the table with the single polygons.
Hope this answers your question.