How to display emails as a post?

It’s very easy use can use wordpress default function wp_insert_post() to create post programtically.
See this for more detail.

Given below is the code that I modified:

<?php
class Email_reader {
    // imap server connection
    public $conn;
    // inbox storage and inbox message count
    private $inbox;
    private $msg_cnt;
    // email login credentials
    private $server="website.com";
    private $user="[email protected]";
    private $pass="PSSWORD";
    private $port   = 993; // adjust according to server settings
    // connect to the server and get the inbox emails
    function __construct() {
        $this->connect();
        $this->inbox();
    }
    // close the server connection
    function close() {
        $this->inbox = array();
        $this->msg_cnt = 0;
        imap_close($this->conn);
    }
    // open the server connection
    // the imap_open function parameters will need to be changed for the particular server
    // these are laid out to connect to a Dreamhost IMAP server
    function connect() {
        $this->conn = imap_open('{'.$this->server.'/notls}', $this->user, $this->pass);
    }
    // move the message to a new folder
    function move($msg_index, $folder="INBOX.Processed") {
        // move on server
        imap_mail_move($this->conn, $msg_index, $folder);
        imap_expunge($this->conn);
        // re-read the inbox
        $this->inbox();
    }
    // get a specific message (1 = first email, 2 = second email, etc.)
    function get($msg_index=NULL) {
        if (count($this->inbox) <= 0) {
            return array();
        }
        elseif ( ! is_null($msg_index) && isset($this->inbox[$msg_index])) {
            return $this->inbox[$msg_index];
        }
        return $this->inbox[0];
    }
    // read the inbox
    function inbox() {
        $this->msg_cnt = imap_num_msg($this->conn);
        $in = array();
        for($i = 1; $i <= $this->msg_cnt; $i++) {
            $in[] = array(
                'index'     => $i,
                'header'    => imap_headerinfo($this->conn, $i),
                'body'      => imap_body($this->conn, $i),
                'structure' => imap_fetchstructure($this->conn, $i)
            );
        }
        $this->inbox = $in;
    }
    function total_msg(){
        return imap_num_msg($this->conn);;
    }
}

$emails = new Email_reader;    
$total =  $emails->total_msg();

    for ($j=1; $j <= $total; $j++) { 
       $mail =  $emails->get($j);

       $post_array = array(        
        'post_content'  => $mail['body'],
        'post_title'    => $mail['header']->subject,
        'post_type'     => 'my-email',
        'post_status'   => 'publish',
        'meta_input'    => array(
                'to'           => $mail['header']->toaddress,
                'email_date'    => $mail['header']->Date,   // add post meta as many as you want
            ),
       );
       wp_insert_post($post_array);
    }
    

I changed your class and added 1 more function to get a total number of emails.
after that, I get email one by one and create a post using wp_insert_post()

Now you can read it using your code:

$args = array(
'post_type' => 'my-email',
);

$emails = new WP_query($args);

while($emails->have_posts()) {
$emails->the_post();

?>
<h3><?php the_title(); ?></h3>

I did not add an Excerpt but you can modify my code according to your need.
Thank you.