The use of '的', please explain to me what is the difference between '自由翱翔' and '自由的翱翔'? Are the literal translations of the two phrases the same?

If they have the same meaning, why would people use '的' in a phrase like this? Would it make a phrase more polite or beautiful?
If translated literally, your two phrases would be fly freely and freedom's flying. In this case (the way I'm interpreting it), the 的 functions as a bit of a possessive. If you meant 地, which in this case would turn the 自由 into an adverb (freely instead of free or freedom), then there isn't much of a difference in meaning. However, the four letter phrase is a common trend in Chinese and is often associated with more poetic and metaphorical statements, so would be more beautiful, yes.

(If you are confused about the finer details of the 的、得、地 grammatical construct, in general 得 is used between adverb and verb when the verb is first, and 地 in the reverse situation, while 的 is usually between adjective and noun.)