I'm not sure that "emoji" is the right word here.

https://www.theguardian.com/technology/2015/feb/06/difference-between-emoji-and-emoticons-explained

However, your smiley is an example of an emoticon and fundamentally it is a short string of typed characters, the classic being the :-) representing a smiling face. Some applications choose to display these as images. When you select a smiley from Thunderbird's menu, it is actually inserting the underlying characters, but choosing to display them as an image.

So, as a generality, if you insert an emoticon, you have to accept that it can and may be shown as the characters, depending on the device and program being used by the viewer. Whilst you might hope that it is shown as an image, this is outside of your control.

The puzzle here is that if you see the image when composing, why do you not when you send a message to yourself? There is a setting in Thunderbird (which I'll need to look up) to control whether or not the conversion from text to image is applied, but IIRC this is specific to plain text and shouldn't make any difference to html-formatted messages. And if you are seeing a smileys menu, you must be composing using html.

OK, I miss spoke. I talking about the Smileys. What do I need to do to be able to see the Smiley when I send an email to myself? It shows on the email before I send it, but not on the email I open it. Thanks!

Can you confirm you are viewing both the original and received messages in Thunderbird?

Yes, I am sending and receiving the emails in Thunderbird.

There are a couple of add-ons that actually add the graphics to the email. See https://addons.mozilla.org/en-US/thunderbird/search/?q=emoj&appver=&platform=

Also check in options > display > formatting that the option to display emoticons as graphics to turned on. It's default is on, but you may have turned it off at some point.