Don't forget to first Download Your Theme (Template). The current theme should always be downloaded and saved to hard disk in case the html gets messed up and you need to put back the code.
Once you are in the HTML, Blogger often hides some of the code and you must click the black arrow in between the row number at the far left and the actual code.
Early on, most templates or themes used the code below. Recently we had to hide the labels on one of the "Simple" themes and the code was still almost identical and it worked successfully.
So carefully following the directions below..
Click in the HTML, then press Ctrl+F on the keyboard to open the "Find" box in the upper right corner of the HTML window. Start typing the following: cond='data:post.labels inside the box and press Enter. Be careful. If Blogger can't find what you've typed and you press Enter, it may start adding blank lines in the HTML, which you don't want.
Hopefully you'll find the code below. When you do, delete ONLY the lines highlighted in red. If your code isn't exactly as below, but you feel it is close, delete the code but place it in a text (Notepad) file and save it in case you need to put it back.
<div class='post-footer-line post-footer-line-2'><span class='post-labels'>
<b:if cond='data:post.labels'>
<data:postLabelsLabel/>
<b:loop values='data:post.labels' var='label'>
<a expr:href='data:label.url' rel='tag'><data:label.name/></a><b:if cond='data:label.isLast != "true"'>,</b:if>
</b:loop>
</b:if>
</span> </div>
Only delete the lines highlighted in red! And your labels will disappear. Cheers!
