Why is a に particle following a na-adjective in the following sentence? I know that the particle has several uses, e.g. direction and location of existence, but I don't know the reason why it is being used in this case.
花が綺麗に咲いています。
The flower is blooming beautifully.
Here, 「綺麗に」 is like English beautifully. You use に for the same reason you use -ly in English.
I wouldn't call に a 'particle' here. What it is depends on which analysis you're using:
In the traditional Japanese grammar learned by students in Japanese schools (学校文法), na-adjectives are called 形容動詞, and they're taught that 形容動詞 inflect depending on how they're used:
綺麗に ← This is an adverbial ending, similar to English -ly. It links the adjective to a following 用言 (an independent inflecting word such as a verb or adjective; basically, a word that can function as a predicator).
綺麗に咲く
綺麗な ← This is an adnominal ending. It links the adjective to a following 体言 (an independent non-inflecting word such as a noun).
綺麗な花
綺麗だ ← This is a conclusive ending. It comes at the end of a clause.
その花は綺麗だ
In your example, 綺麗に is connecting to the following predicate 咲いています. The word 咲く is a 用言, and the に inflectional ending is used to show a relationship to a following 用言, so it's the right choice.
This isn't a complete list of how na-adjectives can be used, by the way. You should be able to find more discussion in your textbook.
Many modern grammars, including many of the grammars taught to second language learners, treat na-adjectives instead as a special kind of noun (a "nominal adjective" or an "adjectival noun"). In this kind of analysis, に is a form of the copula だ, the same だ you find with nouns. The main difference between the two uses of だ is that with nouns, in most cases の is used rather than な.
In this analysis, に is considered a separate word rather than an inflectional ending. It still isn't usually considered a 'particle', though. Why not? Because a particle is a function word that doesn't inflect, and だ changes form.
