This is slightly unintuitive, but in the SQL Developer Export Wizard, uncheck the 'Show schema' option from the Export DDL section, before unticking that whole section.

The insert statements honour that setting even if you exclude the DDL.
With that turned off the generated script looks like:
--------------------------------------------------------
-- File created - Thursday-June-23-2016
--------------------------------------------------------
REM INSERTING into A
SET DEFINE OFF;
Insert into A (NAME,AGE) values ('Tom',1);
Insert into A (NAME,AGE) values ('John',2);
...
If I export again but leave that 'show schema' checkbox on instead I get:
--------------------------------------------------------
-- File created - Thursday-June-23-2016
--------------------------------------------------------
REM INSERTING into STACKOVERFLOW.A
SET DEFINE OFF;
Insert into MYSCHEMA.A (NAME,AGE) values ('Tom',1);
Insert into MYSCHEMA.A (NAME,AGE) values ('John',2);
...