Currently, Thunderbird does not have the automatic XPI installs like Firefox does for add-ons. This presents a problem to many Thunderbird users who have grown accustom to the simple install process in Firefox. Clicking the Install Now button/link will cause Firefox to attempt to install the add-on in itself. This is not a good thing, especially if the add-on is not designed for Firefox. I have been hearing that the future (late 2007/early 2008) Thunderbird 3 is going to make installing add-ons simpler. Until then you will need to follow a few steps. These steps may seem a bit complicated at first but do it a couple times it becomes easier:

1. Once you have located a Thunderbird add-on you want to install, right-click on the Install Now link in the green box.
2. Select Save Link As… from the context menu.
3. IMPORTANT: Some extension authors require you to accept an End User License Agreement (EULA) before you can install the add-on. Before you click Save take a look at the Save as type field:If the Save as type field displays XPInstall Install as it does below, then you can go ahead move on to Step 4.
   If the Save as type field displays Firefox Document as it does below, click Cancel. Click on the Install Now link in the green box. This will take you to another page which will have the author’s EULA. Review the EULA then right-click on the link in the green box. Now the Save as type field should display XPInstall Install and you can move on to Step 4.
   
4. While you can save the download add-on anywhere you like, I find it is easier to save to the Desktop. Select your save location and click the Save button.
5. Minimize Firefox and toggle to Thunderbird (if Thunderbird is not yet open, open it). From the Thunderbird Tools menu, select Add-ons or Extensions (TBird 1.5). The Add-ons or Extensions (TBird 1.5) Manger window will open.
6. From the Desktop or where every you saved the add-on in Step 4, drag the add-on in to the Add-ons or Extensions Manger window and follow the directions.
7. Restart Thunderbird. While you are waiting for Thunderbird to restart you can (optional) send the add-on downloaded in Step 4 to the recycle bin. Once Thunderbird restarts your new add-on is installed.

Edit: There is actually a simple process to install add-ons after all, take a look at the first comment to this post and then: UPDATE: Installing Add-ons in TBird.