(Note: See the last, brief comment first for a short answer. See these other comments if you want a more detailed, longer answer that includes additional factors, examples, and explanation.)
It basically functions as a left-dislocation paired together with a verbless clause. "Left-dislocation" is an unfortunate name because Hebrew is read from right to left (and verbal communication occurs neither from left-to-right nor from right-to-left!). Something like a pre-dislocation or a pre-clausal dislocation would probably be preferable, since it precedes the clause. Anyway, such dislocations can be used to introduce a new, or sometimes complex, idea before making an assertion or comment about it.
For example, if I wanted to introduce someone into a conversation, I could say "Our old neighbor-girl who used to watch the children, she just got married!" I first introduce "Our old neighbor-girl who used to watch the children" into the discourse, and then I can make the comment that I had originally wanted to make with the following clause: "she just got married!" The pronoun "she" refers to the individual who has just been introduced through the dislocated portion of the sentence: "Our old neighbor girl who used to watch the children". It's dislocated outside of the clause, being placed immediately before it: "she just got married". We could also drop the pronoun and include it in the clause: "Our old neighbor-girl who used to watch the children just got married!" Depending on the length and complexity of the sentence, this could get cumbersome and fail to place emphasis on the central point—in this case, what she did or what happened to her: "she just got married!".
The reason that Hebrew does something similar is probably because without an explicit verb indicating a relationship between העוגה הטעימה and בשבילו, it might help clarify, for example, whether the sentence is identifying primarily who the tasty cake is for (or for whom) or what is for him, although there still might be some ambiguity without more context. In other words, the first of the two sentences below might be more ambiguous than the second, especially in writing (since writing lacks, or doesn't explicitly encode, verbal-prosodic information like pitch, volume, stress, etc.):
1) העוגה הטעימה בשבילו
2) העוגה הטעימה היא בשבילו.
It might be even more clear if the benefactor wasn't known (Sarah in this case): העוגה הטאימה היא בשביל שרה.
Also note that Hebrew uses more verbless clauses than English. Duolingo has some notes on this topic, and other users have done a good job treating it as well. The following link doesn't cover everything:
I'll just add, in case it's causing anyone confusion, היא is functioning as a grammatically feminine, singular pronoun, referring to the cake ("it" in English). A wooden translation would be "The tasty cake, it [is] for him." You probably shouldn't regard these as completely, 100% equivalent, however, since Hebrew consistently uses verbless-clause constructions, this would generally be overkill in English if all you wanted to say was "The cake is for him".
That said, dislocations are also found in English and other languages for good reason. For instance, they can streamline the introduction of new ideas/participants into a discourse, help keep less important information in the background and more important information in the foreground, and pack rhetorical punch when needed. (Prescriptivists ought to learn how language actually works before making arbitrary/unwarranted "grammar rules". How's that for a prescription? The best prescriptivists are first descriptivists. I'm talking about the worst offenders here!) :-)
Back to the pronoun היא as it is used here, Duolingo's notes add "We use the personal pronouns kind of like a 'to be' verb"—with "kind of" probably being an important qualifier. It's used in similar contexts and often to communicate similar ideas where a "'to be' verb" is usually more than sufficient, in English.
העוגה הטעימה היא בשבילו
"The tasty cake, it [is] for him." (direct/wooden translation)
"The tasty cake [is] for him." (generally more common and natural in English)
I should have explicitly made the very simple and specific point that AmirLFC made in another comment thread. Concerning the following example, he said: "Without it it can be an unfinished sentence: 'the cake for the children'":
העוגה היא בשביל הילדים The cake, it [is] for the children. (rigid/wooden translation) The cake [is] for the children.
I'd say that's the basic point to take away, whether or not someone finds the nuts and bolts of "left-dislocations" interesting and helpful, or not! That's one simple motivation/reason for this construction (even if other factors come into play). The above expression could be especially ambiguous in certain written contexts without the pronoun היא.
Hi, AniOhevYayin. Another thing to consider is how prepositional phrases can be embedded in (or a part of) larger noun(/substantive/reference)-phrases, which is why AmirLFC said that sentences like "the cake for children" can be ambiguous without a pronoun. For a simple English illustration, notice how the prepositional phrase "for him" is a constituent of the larger noun phrase "the tasty cake for him" (italicized in the sentence below):
"The tasty cake for him is over there."
That phrase might seem slightly odd or contrived, although it's not too hard to imagine a situation in which it might be used. Some additional examples would be "the book on the shelf", "the cup on the table", "the [eye]glasses in the drawer", and "the cat under the car". Contrast this with "the book is on the shelf", "the cup is on the table", the [eye]glasses are in the drawer", and "the cat is under the tree." That's partially why the notes for Duolingo's Hebrew course say, "We use the personal pronouns kind of like a 'to be' verb."
Other factors can come into play as well, which I tried to briefly describe over a year ago. And if anyone scoffs at the word "briefly", recognize that there are articles, sections, chapters, and even entire books and dissertations written on such topics! Probably good bedtime reading for most of us. ;-)
I hope this helps even just a little bit!
Thanks for your response. I'm not interested so much in how English works, but want to get a grip on when the copula is used and not used. In written Hebrew such as BH and rabbinic Hebrew, there are monographs on this topic, but in those cases we are dealing with texts. The discussion of this matter goes back eons, as can be seen in S. R. Driver's brief treatment on p. 268 of his Treatise on the Use of Tenses in Hebrew (first published 1874). Hebrew has nominal clauses at every stage. It's also not necessarily helpful to know how they worked in BH or rabbinic Hebrew, as Israeli Hebrew has its own way of being Hebrew that is indebted to the old ways but not equivalent. Having said that, it would probably be fruitful to see what Muraoka writes in his Modern Hebrew for Biblical Scholars (rev. ed. 1998) regarding nominal clauses / copulas. Thanks for the reminder about the matter of ambiguity in some nominal phrases. I'll try to keep that in mind. תודה רבה
AniOhevYayin, you raise some excellent points!
I was only trying to accomplish a couple things in this entire discussion. First, since I don't know what everyone's background is, I tried to offer a few simple English examples that have analogous constructions in Hebrew (especially the potential ambiguity of some noun/reference-phrase constructions in certain contexts). From your response, I assume that you probably have more experience with Hebrew than many Duolingo users.
Second, I tried to offer some factors that help motivate (or influence) the use of the pronoun in a dislocation construction (or more-or-less as a "copula"). I didn't address your central question, however, about the relative frequency of these grammatical constructions or, more specifically, if "the copula היא is actually preferred here in this sentence." I'm not entirely sure about that either: The absence of the copula could very well be the default. Either way, you're absolutely correct that the Hebrew pronoun is not always necessary. When you do find it, it will depend largely on the (pragmatic) communicative context and the information-structure of the sentence.
For Biblical Hebrew, Van der Merwe and Naude's A Biblical Hebrew Reference Grammar: Second Edition will likely provide one of the most linguistically up-to-date, summary treatments for BH. But like you said that's "not necessarily helpful" for MH. You might find a more up-to-date treatment for MH in the Encyclopedia of Hebrew Language and Linguistics. Thankfully, some of the articles are available online. https://danielomcclellan.wordpress.com/2013/10/06/encyclopedia-of-hebrew-language-and-linguistics-articles-available-online/