Textarea doesn’t support rich text. Those WYSIWYG editors you’re talking about use div with contenteditable="true" attribute set, instead of textarea:
<div contenteditable="true">This content will be editable</div>
And when you edit content in the div, editor puts its HTML code (as plain text with tags) into hidden textarea. You receive value of that hidden textarea when user submits form.
Google for “lightweight wysiwyg editor”. There are lots of them.
To integrate any of those editors (including monsters like TinyMCE) you usually have to just include its JS file to the page and run single line of JS code to replace your textarea with editor.
No, it’s quite easy. Here is how you add TinyMCE:
<textarea>This textarea will be replaced with TinyMCE</textarea>
<script src="//tinymce.cachefly.net/4.2/tinymce.min.js"></script>
<script>tinymce.init({selector:'textarea'});</script>
Just 2 lines