|Let's define some terms here real quick...|
You downloaded a resume template in microsoft word format, when you opened it, you didn't actually open the template itself, you opened an instance of it. Imagine you are a dress maker, and you have a template of a dress, you lay it on some fabric and make all the cuts and lines that the template defines, you then remove the template, and are left with an instance of it... a dress. It's the same concept for document templates in Microsoft Office, except the templates don't make dresses, they make documents, spreadsheets, presentations, etc. Sorry if you knew that already. That idea is hard for some people to understand so I thought I should spell it out.
Now, as far as the problem with words wrapping onto the second page, that is probably happening because there is no space left within the margins on the first page. You should change the margins to 1" all around... top, bottom, left, right. If you haven't changed the margins already, then the right and left are probably at 1.25". Changing them to 1" will give you some space, and the content on the second page should come back up to the first.