How exactly does knowing a variable's probability distribution help you when learning about data? I am an elementary/wanna-be statistician/data scientist from South Korea. I have been studying a variety of theories of mathematical statistics and different probability distributions. (I apologize for bad English if there is any!)
I am wondering how exactly these distributions help a curious data scientist derive any inference or make decisions when building a predictive model.
For example, if a variable of a dataset is estimated to follow Poisson distribution with a specific value of parameter lambda. What can it actually tell you? What kind of inference can I learn? What can it imply when I am looking into a couple of specific data points or areas?
 A: The entirety of statistical inference is based upon assuming that a variable of interest:
1) Has a probability distribution; and 
2) The parameters of the assumed probability distribution can be estimated form the data.
You never know the true probability distribution of a variable (except perhaps in contrived or very unusual circumstances).  If you did, though, you could make a wide variety of claims about real-world events or the randomness of real-world events.
For example, suppose you owned a grocery store and you know that the number of customers per hour followed a Poisson distribution with rate parameter 2.  With this information, you can conclude that, at any given hour of the day you will on average receive 2 customers.  95% of the time, you'll receive between 0 and 5 customers per hour.  If you ran a sale and observed 10 customers arrive per hour, you would have solid evidence that the sale was having its intended affect of increasing customer interest in your store.  In fact, observing 10 customers per hour is almost inconceivable for a Poisson process with mean = 2 (should occur less than 0.0001% of the time), so you would have rather strong evidence.
I'm not entirely certain if this addresses your question, however.  You specifically asked about how knowledge of the distribution could help you with respect to variables in a dataset.  This is kind of an odd question because, as a researcher we are almost never interested in understanding the distribution of data for their own sake.  Most of the time, we look at data to learn what it can tell us about a real-world "data-generating process."  We hypothesize that the data came from some "population-level" process, and we infer the properties of that process from observing data drawn from it.  Typically, the distributions of data are only indirectly useful; they can, for example, help us decide if an assumption of normality (which applies to the hypothesized data-generating process, not to the data themselves) is defensible.
A: Often finding the distribution that fits the data is not an endpoint, but a starting point. By knowing the distribution (whether it is "true" or just an approximation), you can say something about the likelihood of a new observation.
For example, you found your observations fits well to N(0,1). Next you observe x=3.1 (the value you've never observed), whose p(x)<1e-3 according to N(0,1).
Then you could say 1) x=3.1 is an outlier (inference based on N(0,1)), or 2) your N(0,1) "conjecture" was wrong, 3) you need more data to draw some conclusion, etc. 
Without the distribution, you really can say nothing about how likely x=3.1 observation is.
For almost all cases y=f(x)+error. error will prevent you to find out the relation between x and y. Even worse, f is unknown as well. Hence what all you can say is about p(y|X_new), and this is possible only when you "know" the distribution.
(Note that there is no room for probability or distribution in  e.g. y=x+3 case where  everything is certain.)
You come up with a value, and see its likelihood with respect to some distribution. Then you can assess whether this value is "extreme" or not - "unlikely to happen" or not. To make some "judgement", you need distribution.
