In textarea values, the line breaks are marked with the EOL character (/r, /n or /r/n). Those characters do not have any visual impact on HTML (only in the source). What you want is to make them visible by replacing them with <br/>
or <p></p>
Your best bet is to replace those characters when displaying the text, so your user won't see the tags if they edit their comments.
This, with some more options like adding <b></b>
and <i></i>
when text is surrounded by single or double asterisks ( * ) would give you a fairly close formatting to stackoverflow, which does use a textarea unlike you believe.
EDIT: Like suggested by others, there exists WYSIWYG (What You See Is What You Get) editors which means the user sees exactly the result of his comment, and can edit the styles just as if they were in Microsoft Word or similar. This approach might be fine, but it is way overkill for comments. Reddit and stackoverflow allows to see a PREVIEW of your content before posting, but they do not allow to see, while typing, the live results. 2 solutions, both work, you just need to choose the one you think would be best for your users.
Also, when using a simple textarea, you might want to look at markdown, you could provide markdown support (which is very close to how we format our text here) that way some people won't even need to (re)learn how to format their text on your site, as it is the same as other sites / tools.
EDIT2:
Here is a fast/basic implementation of 4 styling applications: single line-break, double line-break to paragraph, double asterisk to make bold, single asterisk to make italic.
$("#editor").on('keyup', function(e) {
var input = $(e.target).val();
input = input.replace(/[\n]{2}/, '</p><p>');
input = input.replace(/[\n]{1}/, '<br/>');
input = input.replace(/\*\*(.+)\*\*/, '<strong>$1</strong>');
input = input.replace(/\*(.+)\*/, '<em>$1</em>');
$('#result').html('<p>' + input + '</p>');
});
#result {
border: 1px solid #777;
background-color: #CCC;
}
#editor{
width:100%;
height:100px;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<textarea id="editor">
</textarea>
<div id="result">
</div>