First thing, ensure that indicator support is installed on your system (packages libindicator7 and libindicator3-7).
This will enable you system to receive indicator messages.
$ sudo apt-get install libindicator7 libindicator3-7
Then on Thunderbird open Tools → Add-ons. There ensure that the Extension named "Messaging Menu and Unity Launcher Integration" is enabled (it should be installed by default).
This will enable Thunderbird to send indicator messages to the system.
Finally restart Thunderbird.
This is how the badge icon appears in the default "Task Manager" (which is your case),
while this is how it appears switching to the "Task Manager Icon Only" applet.
In any case, note that the indicator does not count the number of unread messages, but the new messages instead (which is what you asked for). For example, here I have a lot of unread messages, but only 2 of them are new. That's why the counter indicator shows the number 2.