A family of Microsoft word processing software products for creating web, email, and print documents.
The behavior you want isn't part of the ordinary capabilities of content controls. As in most cases where you want behavior in Word that isn't built in, you need to write macro code. (You can't record this kind of macro, either -- it has to be designed and written by hand.)
For the general idea of a macro that "watches" what the user selects in a content control and then does something special with that information, look at http://gregmaxey.mvps.org/word_tip_pages/content_control_custom_events.html. Greg lays out code for a "monitor" macro that runs whenever the content of any control in the document changes.
You would have to include that code in your template. Then, in addition, you would need to write code inside the monitor macro that says "If the control that changed is this specific dropdown, and if the selected value is 'Other', then insert a rich text content control (below? to the right of?) the dropdown."