Learn Hardcore Japanese: nakigoe de miwakeru. - I distinguish them by their calls.

Not necessarily. By itself, it often feels like a dictionary-form phrase or a headline-style sentence fragment rather than a fully explicit sentence.

What is omitted is usually something like:

鳥を鳴き声で見分ける。
Distinguish birds by their calls.
鳴き声で見分けることができる。
You can tell them apart by their calls.

Japanese often leaves out the object and topic when they are obvious from context. So 鳴き声で見分ける。 is perfectly natural if the listener already knows what is being distinguished.

Why is the particle で used here?

Here, で shows the means, method, or basis used to do something.

So:

鳴き声で = by the cry / by the call / using the call
見分ける = to tell apart / to distinguish

In other words, で answers the question “By what means?”

Compare:

名前で見分ける = tell apart by name
色で見分ける = tell apart by color
音で見分ける = tell apart by sound

So in this sentence, で is very naturally translated as by, from, or using, depending on the English sentence.