It is the construction that corresponds to "relative clauses" in English and it is both comparatively simple to grasp and extraordinarily useful. One major thing it accomplishes is boosting you from simple sentences to more complex sentences.
I have heard that they have long since ceased teaching English grammar in America, so it might be a good idea to introduce relative clauses and relative pronouns a bit:
He is the man. He stole my car. -> He is the man who stole my car.
That is the hospital. I was born there. -> That is the hospital where I was born.
This is the house. I want to live in it. -> This is the house that I want to live in.
Japanese does the same linguistic task in a very simple way....you just have a verb phrase (which may be short, long, simple, or complex) which modifies the following noun. Being able to spot the phrase and recognizing what noun it applies to is key to successfully breaking down longer, more complex Japanese sentences and figuring them out instead of wandering around in them and getting lost.
In the example you asked about, think about what the term すき signifies.....it signifies an interval of time.....that is to say, a duration. And in English what do we use to show duration?......We use "while".
So the reason すきに must be there is because it is the part which indicates the duration of time....the period of time....in which the fall occurred. The bit that comes before it describes or defines the duration....."she wasn't looking". "She wasn't looking" alone only tells what was(n't) going on...the action (or lack of it, in this case). We need the when part as well, in order to combine the two into a single sentence and in this case the when part was a duration. In daily speech, we tend not to make such a fine distinction and we are just as likely to say "when I wasn't looking" as we are "while I wasn't looking". The important thing is to learn to recognize what communicative function things like すきに are fulfilling in a sentence. If you get hung up, ask yourself questions (who, what, where, when, why, which, how) and try to figure out what sort of information a word or phrase must be contributing to the sentence and you'll find that the process of elimination and common sense can go a long way toward helping you figure stuff out. Everything (ideally) has a function and a reason for being there....to communicate some bit of information. Figure out at least what type of information it must be and you're much farther down the road to solving the puzzle.