How can I replicate the pattern on these mittens?

The picture is big enough to count the stitches in each colour and copy the pattern onto graph paper.

This is just the standard Fair Isle patterning. You DO know what Fair Isle is, don't you? If you are just a beginner, I do not suggest that you do this at all. You need to be able to work with yarns in BOTH hands... and that is far beyond a beginner!! I suggest that you work on the basic black ribbed mittens instead.

This is the standard 8-pointed star common in many basic two-colour work (or Fair Isle) patterns. You will find it in any pattern book.... it's very simple to do... when you know how. Nothing is complicated -- all knitting is just working one stitch at a time (in the right order of course).

When in doubt, you should always go to your LYS and ask there... that is what they are for!