This seems to be due to an error in the Java console. In order to deliver an easier installation, we use Java to initiate the LogMeIn install process. To resolve this issue, go to http://www.java.com and download the latest version of Java. You should then be able to run the installer from the website.
If this does not work, download and run the LogMeIn installer directly from here:
You will be prompted to enter your account information during the install.