Learn Russian: Он пробивался вперёд и летел стрелой. - He fought his way forward and flew like an arrow.

This is the "metaphorical instrumental" (or instrumental of comparison). While the instrumental case usually means "with" or "by means of", using it without a preposition can express doing something "like a [noun]". Here, летел стрелой means "flew like an arrow". This pattern is common in Russian idioms and literature, such as выть волком (to howl like a wolf).

How does пробивался translate to "fought his way"?

Let's break down the word-building. The root verb is бить (to hit or beat). The prefix про- often denotes moving "through" an obstacle. Finally, the reflexive particle -ся points the action back at the subject. So, пробиваться literally means "to beat oneself through", which translates smoothly to forcing or fighting one's way forward.

Why is the unidirectional verb летел used here instead of the multidirectional летал?

The word вперёд (forward) gives the action a specific, single direction. Because the subject is moving continuously along one path at this moment, you must use the unidirectional verb of motion лететь (past tense летел). If he were darting around randomly in all directions, you would use летал.

Both verbs (пробивался and летел) are imperfective. What would happen if we used perfective verbs instead?