word choice - When to use 出て来る over 出る

Here is a sentence from the children's story ももたろう, describing when the child appears from within the giant peach.

桃の中から赤ちゃんが出てきました。

My dictionary lists 出て来る as meaning "to come out" with a note that it is a "special class, kuru verb." It also separately lists 出る as capable of meaning "to come forth."

What does using 出て来る accomplish over just using 出る?

I originally read 出て来る as two separate words: the て form of 出る + 来る, translating it as "exited and came." Is that essentially the etymology of 出て来る? Is 出て来る merely an idiom that my dictionary happens to list as a separate word?

Answer

The difference between 出る and 出てくる is that the later is specifically oriented towards the speaker (and if included, which is not always the case, also the hearer).
Compare the next two examples:

 a. 家から出てきた。
    [Someone] came out of the house.
 b. 家を出ていった。
    [Someone] left the house. OR [Someone] left for good.

In (a), we are located outside of the house, and the person inside the house comes out. As a result he is outside with us. Since where we are is "here", the person in the house moves to "here", i.e. comes. That is the meaning of the verb くる. In (b), we are inside the house (=here), and the person leaves, i.e. moves from "here" to some other place that is not "here". And that's the meaning of the verb いく.
But if you just use 出る then the locative specification (the information from where the person comes, and where the person moves to) is simply missing.
However, there is also a temporal use of ～てきた・いった. That is not surprising because linguists know that many languages treat temporal expressions like locative expressions. Most of the examples by Wang Muye have a temporal meaning.

c. 雨が降ってきた。
   It had come to rain.
d. 物価は上がっていくだろう。
   Prices will go up from now on.

Hope that helps.