"random" is often used as if it was a real property of the data under study, where it should be replaced with "uncertain". To give an example, if I ask you what how much money you earned over the past month, and you don't tell me, it is not "random", but just uncertain. However, treating the uncertainty as if it was random allows you to make some useful conclusions.
The noise is not "random" per se, but given that we usually have limited knowledge of how each particular piece of "noise" is generated, assuming that it is random can be useful.
Now whenever you fit a model to some data, you will have residuals from that model. And if treating the "noise" as if it was random was a good idea, then the residuals from the model should be consistent with whatever definition of "randomness" you have used in fitting the model. If they are not, then basically the time series is telling you that the "randomness" you assumed is not a good description of what it actually happening, and it gives you a clue as to what a better description might be.
For example, if I fit a linear relationship for the systematic part, but it is actually quadratic, then the so-called "random" noise will not look random at all, rather it will contain the squared component of the systemtatic part.
To make it even more concrete, suppose that your response $Y$ is a deterministic function of $X$, say $Y=3+2X+X^2$. Now, because you don't know this function you suppose that $Y=\alpha+\beta X + error$, and because you have no reason to doubt the model prior to seeing the data, you assume that the error is just "random noise" (usually $N(0,\sigma^2)$). However, once you actually fit your data and look at the residuals, they will all line up as an exact quadratic function of the residuals. Thus there is a systematic component to the "noise" (in fact the "noise" is entirely systematic). This is basically "Nature" telling "You" that you model is wrong, and gives a clue as to how it could be improved.
The same kind of thing is happening in the time series. you could just replace the model above with $Y_{1}=1,Y_{2}=6,Y_{3}=0.5,Y_{4}=10,Y_{5}=3,Y_{6}=10$ and for $t\geq7$ have $Y_t=10+2 Y_{t-1} -5Y_{t-1}^3 + 2Y_{t-5}$ and the same kind of thing would happen.