Upload the gigpress folder to the /wp-content/plugins/ directory on your web server
Activate the plugin through the 'Plugins' admin menu in WordPress. This will create a new top-level menu called "GigPress".
To list upcoming shows, simply create a new page and put [gigpress_shows] in the page content. This shortcode accepts several parameters - please refer to the documentation for details.
GigPress also comes with a sidebar widget - simply drag the widget into your sidebar, set your options, and save.