Why do irrelevant regressors become statistically significant in large samples? I am trying to better understand statistical significance, effect sizes and the like.
I have a perception (perhaps its wrong) that even irrelevant regressors often become statistically significant in large samples. By irrelevant I mean that there is no subject-matter explanation why the regressor should be related to the dependent variable. Thus irrelevance in this post is a pure subject-matter concept and not a statistical one.
I know that a regressor will be statistically significant given a sufficiently large sample unless the population effect is exactly zero (as discussed here). Hence, an irrelevant regressor that appears statistically significant in a large sample has a non-zero effect size in population. 
Questions:


*

*How come an irrelevant regressor turns out statistically significant?

*Should I look for subject-matter explanation (i.e. try to deny irrelevance) or is this a statistical phenomenon?


This is a continuation of a post where I was trying to clarify how to cure this effect. Meanwhile, here I am asking why it happens in the first place.
 A: In addition to the excellent answers already posted, I will try from another point of view.  All models are approximations, in some sense ... Look at some regression model, and some irrelevant variable is significant. What can explain it?


*

*Maybe it just is not irrelevant, that todays scientific consensus on that matter is just wrong. Apart from that:

*It could be a stand-in or proxy for some omitted variable which is relevant, and which is correlated with the irrelevant variable.

*Some relevant variable, included linearly in the model, could be acting non-linearly, and your irrelevant variable could be a stand-in for that part of the relevant variable. 

*Some interaction between two relevant variables is important, but not included in the model. Your irrelevant variable could be a stand-in for that omitted interaction. 

*The irrelevant variable could just be very highly correlated with some important variable, leading to negatively correlated coefficients. This could be important especially if there are measurement errors in this variables. 

*There could be some observations with very high leverage, leading to strange estimates. 
Surely others ... an important point is that a linear regression model could be a very good approximation with a small sample, only large effects will be significant.  But a larger sample will lead to lower variance, but it cannot reduce bias due to approximations. So with larger samples those inadequacies of the model becomes manifest, and will eventually dominate over variance.      
A: Even if your sample size doesn't approach your population, tiny effects become significant in large samples. This is a consequence of what statistical significance means:

If, in the population from which this sample was taken, the null
  hypothesis was true, is it (XX%) likely that we would get a test
  statistic at least this large in a sample of the size we have?

If your question is something about all people on Earth, then if you take a sample of 1,000,000 (not close to 7,000,000,000) even very tiny effects will be significant, because it's very unlikely to find such test statistics in large samples when the null is true.
There are lots of problems with significance testing, discussed in many places. This is one of them. The "cure" is to look at effect sizes and confidence intervals. 
A: I have borrowed some insight from @QxV to provide an explanation of presence of a population effect even if subject-matter knowledge suggests no such effect.
Suppose there is a population-generating process (PGP) that generates populations with features $X$ and $Y$. The PGP formula is such that $Y$ and $X$ are independent random variables. Due to randomness, any finite-length realization vectors $y_{realized}$ and $x_{realized}$ have zero probability of exact uncorrelatedness, i.e. $P(y_{realized} \perp x_{realized})=0$. If so, with probability one there is a population effect. That is how effects come about in population.
Once a population effect exists, it is a matter of sample size when we will detect it in the sample and when it will become statistically significant.
A: Questions:

How come an irrelevant regressor turn out statistically significant?

I think it's helpful to think about what happens as your sample size approaches the population itself. Significance testing is meant to give you an idea of whether not an effect exists in the population. This is the reason why when working with census data (that surveys the population), significance testing is meaningless (because, what are you trying to generalize to?).
With that in mind, what does "an effect in the population" mean? It simply means any relationship between variables in the population, regardless of how small (be it a 1-point or 1-person difference), even if that relationship is due to chance and randomness in the universe. 
Thus, as your sample approaches the size of the population, significance tests become less and less meaningful because any difference will be "statistically significant". What you would be more interested in then is effect size - which is analogous to "practically significant".

Should I look for subject-matter explanation (i.e. try to deny irrelevance) or is this a statistical phenomenon?

It's a phenomenon - you should look at effect sizes.
A: No. Irrelevant regressors do not become statistically significant as sample size increases. Try the following code in R.
y <- rnorm(10000000)
x <- rnorm(10000000)
summary(lm(y~x))
