I've always loved this prompt. It's so non-intuitive to the average user. Most click yes when actually have to click No for it work.
Anyhow to disable this in IE8:
(I believe the menus are hidden by default so you will have to press the ALT key.)
Tools>Internet Options>Security
Select the Security tab
Click the Custom Level button
In the Miscellaneous Section, Look for ?Display mixed content? and change it to Enabled
I think LogMeIn updated their site to get around this HTTPS mess. I'd download and run
CCleaner (Piriform) to make sure your browser isn't using cached http: version of the files.