Why should random numbers come from a distribution? Who decided that random numbers have to come from a distribution? Can random numbers come from no distribution, or an infinite mixture of distributions? I am looking for a conceptual answer to this question not a mathematically rigorous one.
 A: We often think of random numbers "coming from", or more aptly, "belonging to" a distribution, because this is useful, and this is how we generate them computationally.
However, it is perhaps more useful to think of random numbers as an endemic property of the universe, and the distribution is what we use to describe them.
For example, suppose you wrote some numbers on some balls and put them in a box. Whenever you needed a number, you would shake the box, and pick out a ball with a number on it. Where do those numbers come from?
They come from the box!
Now, we might say, "but we can describe a distribution of those numbers." Yes, this is true, but the numbers still aren't coming from the distribution. They are coming from the box. The distribution, rather, is a way of abstractly describing the box.
We like distributions because they have nice mathematical properties, and they work well with a notion of randomness that we observe in the universe. It may be better to describe distributions as models of phenomena with no determinism, but I'm not willing to make that philosophical leap.
In the end, if you have randomness, you have a distribution. But the distribution does not determine the randomness; the distribution is a consequence of the randomness.
A: The reason that 'all random numbers have to come from a distribution' is the same as the reason that 'all maps from $\mathbb{R}\to\mathbb{R}$ have to come fron a function'; namely, that one defines the term itself (be it function or random — more properly probability — distribution) as any entity that satisfies all of the hypotheses one would want to be associated for such a thing.  For instance, for probability over the integers $\mathbb{Z}$, one can define a probability distribution as any function $f(n):\mathbb{Z}\mapsto\mathbb{R}$ with $\sum_{n=-\infty}^\infty f(n) = 1$ and $f(n)\geq 0\ \forall n$.  Note that this subsumes the usual notion of, e.g., rolling a single die, where $f(n) = \frac16, 1\leq n\leq 6$ and $f(n)=0$ otherwise; but it also includes much more complicated distributions such as the Poisson distribution, and even uncomputable distributions such as the Solomonoff's 'universal prior' based on the Komolgorov complexity of the number.
This then immediately gives that, e.g., the (appropriately weighted) sum of two distributions or even infinitely many distributions is still 'a' distribution over the appropriate space; for instance, if you have a set of $\mathbb{Z}$ many probability distributions over $\mathbb{Z}$, $\{f_n(t)\ |\  -\infty\lt n\lt\infty\}$, and you have a single 'global' distribution $g(t)$, then the distribution obtained by using $g$ to choose an $f_n$ and then choosing a number from that $f_n$ is still a single distribution; its probability function is $f_g(t) = \sum_{n=-\infty}^\infty g(n)f_n(t)$.  (You should take the time to convince yourself that this defines a proper probability distribution - for instance, that the sum converges for every $t$!)
