Meaning of 的 in 在蓝天白云的 衬托下，阳光的照射下
衬托 = set off (by contrast)
在蓝天白云的衬托下，= under the blue sky and white cloud's contrast
阳光的照射下. = under the sunshine.
An object here is omitted

A complete sentence:
[红衣少女] 在蓝天白云的衬托下，阳光的照射下，[看上去有如女神般美丽]
[The girl in red] under the blue sky and white cloud's contrast, under the sunlight, [looks as beautiful as a goddess]

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What does 的 mean in this sentence?

的 is a possess marker. 蓝天白云 is a noun phrase, and it possesses the noun 衬托 (contrast)
在蓝天白云[的]衬托 = Blue sky and white cloud['s] contrast = the contrast of blue sky and white clouds (in the back ground)
If 衬托 is used as a verb, the sentence would become 被蓝天白云衬托著 (set off by blue sky and white cloud)