I've read up on the causes of random error and noticed insufficient data to be one of the causes of random error. However, I'm confused as to why insufficient data could possibly lead to random errors. Could someone provide some context for why this might be?
More datapoints (if everything is done correctly) lead to more trustworthy results. With less datapoints, the possible (statistical) error grows, until at one point, when you only have as many datapoints as you have parameters, it looks like you have no error at all. (That is of course stupid, you should always have derived the error range of the individual data points.)
Now what is "insufficient"? Is that when the statistical error grows too large for your feeling? Or when you only have two points to draw a straight line through? Or when you only have two points to draw a parabola? That's your prerogative.
(note that I have also above used the word "error" in the same sloppy way your books seem to do. The "errors" are in your data, and they have a systematic component (hopefully small) and a distribution. The derived values (e.g. the slope of a linear regression) have a confidence interval.)
When making a series of measurements we implicitly assume that there is a an 'error', or better, an uncertainty associated with the measurement and that this uncertainty is random and not so influenced by that of any of the others, i.e. the errors and random and independent of one another.
In virtually every case, no matter what the noise on a measurement is, if a sufficiently large number are made then the histogram of the measurements will follow a gaussian or normal distribution, which is a 'bell shaped' curve. The peak of the curve is then taken as the average or mean value of the measurement. Depending upon how wide the bell shaped curve is allows us to put an error limit on the measurement. (The theorem supporting this is called the Central Limit Theorem)
However, we cannot always make a huge number of measurements and so if we make only a few then they could fall anywhere on this distribution; they could be exactly on the mean value or way off this value, one just does not know. In this case we have to use make estimate of the true value by averaging the measurements and by assuming that the noise is random and gaussian (normally) distributed we can estimate the 'error', called the standard deviation.
Usually the error is stated as a mean value plus or minus the standard deviation ($\pm 1 \sigma$) for example $23.4 \pm 0.3$ which means that $\approx 68 $ % of the time the measurement falls within the limits $23.7$ and $23.1 $ but by random chance has a $32 $ % of being outside these. This is not very convincing so it is better to try to obtain $\pm 2 \sigma$ which is $\approx 94 $ % chance of the measurement value being within the range, and by random chance $6$% of being outside this. Of course it is up to you to decide what is appropriate.
An additional point to bear in mind is that the signal to noise ratio is reduced as $\sqrt n$ for n measurements. Thus to improve the signal to noise ratio of a measurement can involve a lot of extra work.
So to answer your question directly; the errors are there all the time, insufficient data does not make them greater or smaller but instead does not average them away to a smaller value.