As you have found, you need to add the shortcode [search_listings] to the body of a page to see the search results and, with the [search_form] shortcode embedded in the same page, you will be able to filter the results using a search form.
It looks like there may be an issue with the Placester: Listings Quick Search widget which we are investigating. Meanwhile, you can create your own search widgets to use in your theme's sidebars:
From the admin menu, select 'Placester'->'Shortcodes/Widgets' and then click 'Add New'. Give the widget a title and then select the type 'Search Form'. If you look at the space between the title box and the 'Placester Widgets' box you should see that a shortcode has been generated. You can copy that and paste it into a Text widget in the 'Appearance'->'Widgets' screen to generate a search form in the sidebar.
On the same admin page you can modify the fields and layout of the form, and there is a preview pane that gives you an idea of how the form will appear.
We are planning an update to the Placester shortcode admin page to make it easier to select the fields that you want to appear in the form and style them as needed.
Please let us know if you need further help.