WordPress can only use a page as the front page (Settings > Reading), and the /places/ listings page is a post type archive, not a page. GeoDirectory has no setting to put that archive on the front page. You can get the same result in one of two ways.

Build a home page with the same widgets

The /places/ archive uses GD > Loop, which only works on the archive template. On an ordinary page, use GD > Listings in its place.

1. Create a new page.
2. Add a GD > Search if you want a search bar.
3. Add a GD > Map and set Filter by tags to .geodir-listings. With a CSS class in that setting, the map only shows markers for the listings on the page, the same way the archive map does.
4. Add a GD > Listings for your post type with the sort order, number of listings and pagination you want.
5. Go to Settings > Reading, choose A static page and select your new page as the homepage.

As shortcodes, a page that lists places A to Z, ten per page, with a matching map looks like this:

[gd_map width="100%" height="425px" tags=".geodir-listings"]
[gd_listings post_type="gd_place" sort_by="az" post_limit="10" with_pagination="1"]

More on the map setting: Showing a filtered list of listings in a special map.

Redirect the homepage to /places/

If you want visitors to land on the archive itself, redirect the front page with a redirect plugin or this snippet:

add_action( 'template_redirect', function() {
	if ( is_front_page() ) {
		wp_safe_redirect( home_url( '/places/' ), 302 );
		exit;
	}
} );

Change /places/ if your post type uses a different slug. With a redirect, your home URL has no content of its own, so the first option is usually better for search engines.