Moving this discussion to the Premiere Elements forum.
I don't think you can remove it - it's a feature of the trial version (see the limitations of the trial version published by Adobe). It goes when you buy it.
Welcome to the forum.
As has been stated, as long as the copy of PrE is the "trial," that watermark will appear, and be embedded in any output file.
When you do purchase the program, and plug-in your S/N to convert from trial to full-paid, you can remove that watermark, for subsequent output, but be sure to Delete any Render Files, as mentioned in that FAQ Entry, posted by John T.. That is an important step, and remember, any files output FROM the trial, WILL always have that watermark.