The problem is that the SSIS package is being executed under the NT credentials of the account running the SQL Server Agent service (likely to be either your local PC's "local system" or "network service" account), and is attempting to connect to the remote SQL instance with windows integrated security. This fails because the remote SQL server cannot authenticate the service account as permitted to connect.
When you developed it, the package was connecting to the remote SQL server using your NT credentials (a domain account which the remote SQL server could authenticate).
The quickest fix will be to change the service account under which your SQL Server Agent service runs to a domain account with rights to connect to the remote SQL server. To prove that this is the issue you could use your own domain account for this, although this isn't really a long term solution.