I tried installing LastPass in Safari, but nothing seems to happen.
Please first check to make sure LastPass installed into Safari via Safari >> Preferences >> Extensions Tab. LastPass should be listed and a checkmark visible in the box next to Enable LastPass.
If LastPass is listed and enabled, please check to the left of the URL bar for a grayed-out * icon. Click on this, and log in to LastPass. The icon will then turn black to indicate you have logged in.