The whole clause is a ‘complex’ sentence. The main (independent) clause is I can see by my watch, without taking my hand from the left grip of the cycle; the subordinate (dependent) noun clause is that it is eight-thirty in the morning.
Without taking my hand from the left grip of the cycle is a prepositional phrase functioning as an adverbial modifier of the main verb ‘see’ in the main clause.
The commas, I believe, were used to set off the nonrestrictive (nonessential) element (without taking my hand from the left grip of the cycle) from the rest of the sentence. Without that element, the Author thinks the reader still can understand the full meaning of the word or words it modifies, as can be seen in I can see by my watch that it is eight-thirty in the morning.
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