Intuitively, why should a random variable have a distribution? When we think about random variables or random processes, why do we make the a priori assumption that a particular realization had to come from a distribution? Why do we even have the concept of distribution? Is there a way for the concept of randomness to exist without the notion of distribution?
 A: When dealing with random events, your goal is make intelligent guesses / predictions, otherwise all you can do is say "This is random, I don't know what to do" and go home crying.
Philosophically speaking,  Distributions and Random Variables are ways to model random events given our prior knowledge (Bayesian approach) or the frequency of past events (Frequentist approach).
And you know what, it actually works...
For example, if you work in the logistic department in a hospital and need to buy diapers for the new born babies, Empirical data shows that modelling the weight of newborns as a normal distribution makes sense, and seem to work quite well.
A: In measure theory and in probability, a random variable is a measurable function
\begin{align*} X: &\Omega \longrightarrow \mathbb{R}\\&\omega\longrightarrow X(\omega)\end{align*} from a set of possible outcomes $\Omega$ to the set $\mathbb{R}$. The set $\Omega$ is a probability space, i.e.,  $(\Omega, \mathcal{F}, P)$ corresponds to a (usually Borel) $\sigma$-algebra $\mathcal{F}$ and a probability $P$ over the elements of $\mathcal{F}$. For instance, $(\Omega, \mathcal{F}, P)=([0,1],\mathcal{B}([0,1]),\mathcal{U})$ where $\mathcal{U}$ is the uniform measure can produce all real valued random variable with distribution $F_X$ as $$X(\omega)=F_X^{-1}(\omega)$$That we do not know the distribution of a random variable about to be observed or already observed does not impact the very existence of this distribution. 

Addendum: There is a philosophical debate in the Philosophy of Sciences
  about the very notion of randomness, that is whether or not it
  exists at all, but I do not think this pertains to your question.

Statistics is concerned with the approximation of this distribution within a certain family of distributions, either parametric, i.e., parameterised by a finite dimensional parameter $\theta$,$$X\sim f_\theta(x)$$ or non-parametric, which means using a family of approximations, e.g.,$$X\sim\frac{1}{n}\sum_{i=1}^n\mathcal{N}_p(x_i,\Sigma)$$It is clearly aknowledged that those are approximations in most settings and the ``truth" about the genuine distribution cannot be established with a finite set of observations.
