There are one of two ways to fix the problem. Try this one first:
1) In your Joomla Administrator backend, go to Site > Global Configuration.
2) Find where it says "Default Editor." Change it to one of the other options. It's been a while, so I forget which editor will work (I think it begins with an "n", though). You'll likely just have to try them all.
3) When you pick a different one, click "Save." Then go back to the article where you're trying to embed the Livestream channel.
4) You'll see the HTML code of your article, and can embed it into there without the code stripping.
Option #2 - Now the newest version of Joomla I noticed does not allow this type of editor anymore. So here is what you'll have to do.
1) From your administrator backend, go to "Menus" and then pick the menu in which you're trying to place a link to your video player. Mine is the Main Menu.
2) Create a new menu item. (Click on the "New" or "Add New Menu Item" button).
3) Click on "Menu Item Type" to select a type of menu item to create.
4) Select Wrapper > iframe Wrapper.
5) After naming your new menu item (I called mine "Videos"), go to Basic Options > URL. You are going to put the URL to your video player in here.
Here is where you get that URL:
A) Find your LiveStream embed code (the code Livestream gives you to embed your channel).
B) Look for the first part of the code that reads iframe src="A URL"
C) Copy the URL that is between the two quotes there ("here"). For example, mine is http://cdn.livestream.com/embed/miss...512&height=480
D) Don't copy anything but the URL between the two quotes. This is what you will paste inside Basic Options > URL on the Joomla New Menu Item page.
Lastly, look for "Scroll bar parameters" on the Joomla New Menu Item page.
Change "Scroll Bars" to "no." Then type in the Width and Height of your LiveStream embedded channel.
For example, if mine is http://cdn.livestream.com/embed/miss...512&height=480
...then I would type Width = 512 and Height = 480 into the Scroll Bar Parameters section of the Joomla New Menu Item Page.
Save it, and you're done.
Now, there is one catch with this method: It won't embed it into a preexisting article. It will only embed it on it's own page. But it should work.