Sample code to illustrate how to use the Updater class to activate a license key.
<?php

function edd_sample_activate_license() {

	// listen for our activate button to be clicked
	if( isset( $_POST['edd_license_activate'] ) ) {

		// run a quick security check
	 	if( ! check_admin_referer( 'edd_sample_nonce', 'edd_sample_nonce' ) )
			return; // get out if we didn't click the Activate button

		// retrieve the license from the database
		$license = trim( get_option( 'edd_sample_license_key' ) );


		// data to send in our API request
		$api_params = array(
			'edd_action' => 'activate_license',
			'license'    => $license,
			'item_name'  => urlencode( EDD_SAMPLE_ITEM_NAME ), // the name of our product in EDD
			'url'        => home_url()
		);

		// Call the custom API.
		$response = wp_remote_post( EDD_SAMPLE_STORE_URL, array( 'timeout' => 15, 'sslverify' => false, 'body' => $api_params ) );

		// make sure the response came back okay
		if ( is_wp_error( $response ) || 200 !== wp_remote_retrieve_response_code( $response ) ) {

			if ( is_wp_error( $response ) ) {
				$message = $response->get_error_message();
			} else {
				$message = __( 'An error occurred, please try again.' );
			}

		} else {

			$license_data = json_decode( wp_remote_retrieve_body( $response ) );

			if ( false === $license_data->success ) {

				switch( $license_data->error ) {

					case 'expired' :

						$message = sprintf(
							__( 'Your license key expired on %s.' ),
							date_i18n( get_option( 'date_format' ), strtotime( $license_data->expires, current_time( 'timestamp' ) ) )
						);
						break;

					case 'disabled' :
					case 'revoked' :

						$message = __( 'Your license key has been disabled.' );
						break;

					case 'missing' :

						$message = __( 'Invalid license.' );
						break;

					case 'invalid' :
					case 'site_inactive' :

						$message = __( 'Your license is not active for this URL.' );
						break;

					case 'item_name_mismatch' :

						$message = sprintf( __( 'This appears to be an invalid license key for %s.' ), EDD_SAMPLE_ITEM_NAME );
						break;

					case 'no_activations_left':

						$message = __( 'Your license key has reached its activation limit.' );
						break;

					default :

						$message = __( 'An error occurred, please try again.' );
						break;
				}

			}

		}

		// Check if anything passed on a message constituting a failure
		if ( ! empty( $message ) ) {
			$base_url = admin_url( 'plugins.php?page=' . EDD_SAMPLE_PLUGIN_LICENSE_PAGE );
			$redirect = add_query_arg( array( 'sl_activation' => 'false', 'message' => urlencode( $message ) ), $base_url );

			wp_redirect( $redirect );
			exit();
		}

		// $license_data->license will be either "valid" or "invalid"

		update_option( 'edd_sample_license_status', $license_data->license );
		wp_redirect( admin_url( 'plugins.php?page=' . EDD_SAMPLE_PLUGIN_LICENSE_PAGE ) );
		exit();
	}
}
add_action('admin_init', 'edd_sample_activate_license');