Without seeing a lot more of the project, it's really hard to tell what's wrong with it. Could you link us to the template on their site?
It is usually more useful to post a screenshot of your project in AE with showing the layers giving you problems with all of the modified properties of that layer revealed. Most of the time a text layer in a template project is in a nested comp (pre-comp) so you may need more than one screenshot. Select the text layer first, press the U key twice to reveal all modified properties of the layer, then if you are on a Windows machine press PrintScreen and simply paste (Ctrl/Cmnd + v) into the reply field of this forum. If you are on a Mac then press Shift+ Ctrl/Cmnd + 3 and drag the screenshot from the desktop to the reply field.
Repeat this with the other comps that contain this text layer.
My suspicion is that the text comp is nested inside another layer that contains a box around it so you'll have to either change the box or the font size to make it fit.
The link you provided does give us access to a Free template but it doesn't tell us what you have done to it and I don't have time to download, open and try and guess.
Thanks everyone, Rick had the right answer, I found the box and expanded it to show all the text.