You are using single quotation in single quotation which is not possible, use one of the below approch,
Approch 1 (' \' ')
:
'post_alert_user' => esc_html__('Thank you for your submission at ', 'usp-pro') .$user['admin_name'] . esc_html__('! If submissions require approval, you\'ll receive an email once it\'s approved.', 'usp-pro'),
Approch 2 (" ' ")
:
'post_alert_user' => esc_html__('Thank you for your submission at ', 'usp-pro') .$user['admin_name'] . esc_html__("! If submissions require approval, you'll receive an email once it's approved.", 'usp-pro'),