If you only have one toolbar item and one popover (and never plan to add more), then it's just one line. Assuming you've already assigned the popover to the toolbar item in Extension Builder, you can just use:
safari.extension.toolbarItems[0].showPopover();
But if you have more than one popover and (potentially) more than one toolbar item, here's a generalized function to open a popover, specified by its identifier, under the specified toolbar item in the active browser window:
function showPopover(popoverId, toolbarItemId) {
var toolbarItem = safari.extension.toolbarItems.filter(function (tbi) {
return tbi.identifier == toolbarItemId && tbi.browserWindow == safari.application.activeBrowserWindow;
})[0];
var popover = safari.extension.popovers.filter(function (po) {
return po.identifier == popoverId;
})[0];
toolbarItem.popover = popover;
toolbarItem.showPopover();
}