how to understand "turn into"?

I think your confusion is justified here, with the word "into" used twice in very different ways.

The first "into" could (and maybe should) be changed to "on". The word "feet" is understood, but only if you know something about snowshoes. I slip my feet into my snowshoes -- or -- I slip on my snowshoes.

The second "into" means that the writer has physically turned his body from the road, and to the bush. "Enter" could be substituted: I entered the bush.

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