Where by "why" I do not mean "list of use cases for randomness." If one has a quantitative question Q about topic X, it does not seem intuitive that values that by definition have absolutely nothing whatsoever to do with topic X would be so crucial in answering Q. And yet, a source of randomness is absolutely essential to most statistical operations. Probably the most common such example is the need to obtain random samples for inferential methods, however the need for randomness is so ubiquitous it seems almost fantastical. So, by "why" I am perhaps looking for something information-theoretical, or even possibly philosophical?
To create a context around Q and X, suppose a statistics student has an assignment to estimate the average height of students at the local college. They come to you and ask about where they should start. Naturally, one place to start involves collecting a random sample of students, so you say: "well, first, you'll need to obtain a Geiger counter." (with the intent that the student use it for generating random numbers). The student adopts a confused expression.
@Tim has suggested that requiring a Geiger counter for this task "at face value is rather ridiculous", which is precisely my point. Other options for obtaining randomness to help with the experiment include lottery balls, atmospheric noise, or repeatedly squaring your zip code: none of which have anything whatsoever to do with measuring heights. In fact, absent a priori knowledge about the population distribution of heights, involving students' heights in the method for obtaining your sample is probably a bad plan. An important part of estimating the average height of the students is finding an activity that is totally unrelated to the heights of students in any way, and obtaining measurements of that thing.
Is there some intuitive explanation to illustrate why a Geiger counter would be so immensely useful for someone interested in investigating the average height of students at the local college?