You have to select "To download another item, view the full list of downloads for your platform." from the LastPass download page.
This will take you to another page where you can download the plugin.
If you select the "No Binary Features" option, you will be redirected to Chrome Web Store and add the plugin.
If you select the "Binary Features" option. you will download a .crx file. Chrome will tell you that it can not add the extension.
What you have to do is open the "Extensions" tab in Chrome and then drag and drop this file on it from the folder you have downloaded it.
This will install the extension.
About the "binary" component:
The "binary" component of the browser plugin controls a few LastPass features:
1. Autologoff on browser close (LastPass Icon > Preferences)
2. Autologoff on browser idle (LastPass Icon > Preferences)
3. Share login state between browsers (LastPass Icon > Preferences > Advanced)