Hmmm, depends on which regional / district variants of English one is used to. In certain parts of Britain that might not sound completely off the radar. There are a lot of entrenched regional "accents" which routinely use grammatical structures that aren't "Queen's English" :-)
Let's for a moment remove the third word
היא שברה את הצלחת
Can't get more basic than that. This is the simplest subject-verb-object construction: She broke the plate.
So the odd thing about this is the indirect object לי
I can't really think out a good way to say this in English. It is not the same as "she broke my plate", because the Hebrew sentence stresses that her action is something that was done TO ME, not that I own the plate. But semantically there is not much difference.
Can you clarify what you mean by "done to you"? How is it different from someone breaking your plate by adding li? Do you mean something done "for you"? If it's hard to describe, can you give an apples to apples comparison (if possible) where it would be needed (li) and where it wouldn't be needed? Thanks.
The word לי in this sentence has the role of adverb, not adjective. As such, it does not modify the plate, but the breaking.
What it says about the breaking is that it was something that was done to me, something that happened to me, something that I experienced.
So why is this breaking of the plate related to me? The sentence is vague about that. Maybe it was done to me because the plate belongs to me and now my plate is broken. Alternatively, maybe it's not my plate at all, but she broke it over my head. The sentence fits both, but the latter option is less likely, because the sentence does not give a hint that this is what happened. So consider the following three sentences:
היא שברה את הצלחת
היא שברה את הצלחת שלי
היא שברה לי את הצלחת
They all have her breaking the plate. The first does not imply I'm connected to this in any way. The second describes the plate with an adjective -- the broken plate belongs to me. The third describes the action -- the breaking of the plate was something that was done to me, for me, to spite me, or some other connection to me.
It depends on what you want to say.