The first step to exporting form submissions is to create the export. After an export is created, you can skip to running the export.
You can do this by navigating to Site -> Exports -> New +
Once you're done configuring the export, click Save, then the X in the top right to close the dialog.
Exporting is always done from Site -> Exports.
TIP: To download the up-to-date data, always begin with #1 (Running the export) first. If you skip #1/Running the export, you'll just download the last version of the data you ran.