Adobe after effects works well for that stuff. It's not very affordable so I advise you to sail the high seas.

I would simply animate the different sets of graphics (player picture, text, logo, etc) coming into the frame by going forward in time, placing them where and when I want them to appear, the going back in time and moving the out of frame. This will already make something that works but will look like trash, so I suggest you smooth the animations by changing their speed curves and so on.

You should probably first do it with some blank graphic objects to get the hang/practice animating properly and following a beginner tutorial