You might take a look at ember-css-transitions, and even if you don’t install it you could borrow from its implementation.
Use of {{did-insert}} (and probably also did-update if you want it to change if you supply the component with more text) should work here; can you put up a link to a sample repo (or an Ember Twiddle) showing how you can’t get it to work? I would assume a very small variant on the jQuery just using (a) JavaScript addition and/or removal of the classes and (b) a CSS transition should do the trick just fine.