I'm not much more of an expert at this than you, but I recently was fiddling with the "Posted by Admin at [date}" thing myself at my blog, so I might be able to help -- assuming that your blog's theme is anything like mine.
This will involve some delicate "hand coding," so be careful!
In your blog's Dashboard, click on "Appearance" on the left sidebar, then "Editor." This takes you to a page where you can directly edit your blog's behind-the-scenes "php" and "CSS" files.
In the list of files on the right side of the page, choose either "Single Post" or (if you have this choice) "Page" or something like that.
Then in the edit window that appears, you will see a bunch of complicated code. What you need to do is delete the specific code that causes the "by author" and "at date" to appear.
In my file, the code looks like this:
<div class="date">Posted by <?php the_author() ?> at <?php the_time('g:i a'); ?>, <?php the_time('l') ?>, <?php the_time('F') ?> <?php the_time('j') ?>, <?php the_time('Y') ?></div>
If yours looks similar, then try highlighting that exact snippet of code, hit "backspace," and it will be erased. Then click on "Update file" below the edit window. Then go to your blog's "Bios" page and refresh it to see if it worked!
Before doing this, save (in a separate document) a pristine untouched pre-edit version of the code in your "Single Post" (or whichever) edit window. If your editing experiments go horribly awry and you can't remember how or where to restore the code you deleted, then you can simply copy the entire original code from your backup document once again and put it back in the edit window, "Update file" once again, and everything will be restored to the way it was originally.
When deleting the code, try to carefully delete the entirety of distinct code statements -- in the example given above, delete everything starting with (and including)
<div class="date"> up through (and including)
</div> at the end. If you leave half a code command -- like if you go into a "div" or a "class" or whatever but delete the part that closes the command, or vice versa -- then you are likely to mess things up and obviously it won't work.
I can't be more specific, because I don't have your theme, and each theme's code is slightly different, and also because I'm a non-expert as well, but editing the code this way definitely would solve your problem if you did it precisely. Perhaps someone else can add to my suggestion with more exact instructions. Alternately, if you see what I'm talking about and want to take the plunge, then go for it (after taking the "Important" precautions noted above)!