How can I interpret 不要为你的美丽伤了我?
This is quite typical of language in mainland-Chinese public service announcement type signs, which tend to be quite &quot;cutesy&quot; in tone. Stuff like personifying inanimate objects (such as flowers) and writing it from the object's point of view fits with this tone.
The literal meaning is:

don't hurt me for [the sake of] your beauty

Which I suppose is hinting that the reason you might trample the grass is to take an attractive selfie/photo.
Edit: From the photo you attached, it's more likely referring to the flowers than the grass (in other words, don't harm the flowers by picking them).