In your example, "looking through the window" is an adverbial phrase defining the verb "saw." By the way-- you should always capitalize the subject pronoun "I", and you should also insert a comma after "window."

In the sentence "Looking through the window, I saw him traverse the garden", the phrase "Looking through the window" acts as a participial phrase. Explanation: Participial phrases begin with a present participle (a verb ending in -ing) or a past participle (typically ending in -ed or -en) and function as adjectives or adverbial phrases in a sentence. Here, "Looking" is the present participle of the verb look, and it describes the action being performed by the subject of the main clause, I. In this context, the participial phrase is adverbial and modifies the verb "saw", explaining how or under what circumstances the action occurred. In this case, it specifies that the speaker saw someone traverse the garden while they were looking through the window. So, the phrase "Looking through the window" provides additional information about the conditions under which the speaker saw the action happen.

to look through - phrasal verb