The animation that shows the invitation you designed coming out of the envelope will take place one time for each guest. If you have viewed your invitation and did not see this animation the most common cause is the invitation or save the date has already been opened. Once an envelope has been "opened" the next time you view the invitation it will be "opened".
Another possibility is that Party City is unable to load the animation in a timely manner and the animation is skipped. We do this for slower internet connections or older web browsers so your guests are not waiting too long for their invitation to open.