It sounds like you're pasting the html code directly in to the HTML editor while it's in the default/normal view. To properly insert that HTML code, click the little "HTML" button in the HTML editor to switch it to show the html code for the area you're editing. I suggest you add a few lines to the end of whatever code is in there, followed by a few "<BR />"s (adds a new line).. Then paste in the html iframe code you got from google.
See the HTML editor is actually generating HTML code behind the scenes as you use it. Then it saves that code to the server. So everything that you enter is interpreted and is converted in to a guess at what the equivalant HTML is.
If you dump HTML code in to the normal view, then that code is actually converted so that it'll show to users the way you entered it - unless you dump it in from the HTML view, then they'll see what the code actually does.
The link code that Helen refers to could also be an option - the iframe approach will embed the document within your page in moodle. Also there are parameters in the iframe "src" attribute you can edit.. Mainly, look for the one that reads "chrome", try switching that between "true" and "false" - you'll likely prefer one over the other.