We don't know what happened between the samples.
A crude example is to consider a 'glitch' that happened to fall between adjacent samples. Since we don't measure it, we have no way of knowing the glitch was there at all.
In a less obvious case, we might have signal components that are varying rapidly in between samples. Again, we could not track these rapid inter-sample variations.
We must sample fast enough to see the most rapid changes in the signal.
Sometimes we may have some a priori knowledge of the signal, or be able to make some assumptions about how the signal behaves in between samples.
If we do not sample fast enough, we cannot track completely the most rapid changes in the signal.
Some higher frequencies can be incorrectly interpreted as lower ones.
In the diagram, the high frequency signal is sampled just under twice every cycle. The result is, that each sample is taken at a slightly later part of the cycle. If we draw a smooth connecting line between the samples, the resulting curve looks like a lower frequency. This is called 'aliasing' because one frequency looks like another.
Note that the problem of aliasing is that we cannot tell which frequency we have - a high frequency looks like a low one so we cannot tell the two apart. But sometimes we may have some a priori knowledge of the signal, or be able to make some assumptions about how the signal behaves in between samples, that will allow us to tell unambiguously what we have.
Nyquist showed that to distinguish unambiguously between all signal frequency components we must sample faster than twice the frequency of the highest frequency component.
In the diagram, the high frequency signal is sampled twice every cycle. If we draw a smooth connecting line between the samples, the resulting curve looks like the original signal. But if the samples happened to fall at the zero crossings, we would see no signal at all - this is why the sampling theorem demands we sample faster than twice the highest signal frequency.
This avoids aliasing.
The highest signal frequency allowed for a given sample rate is called the Nyquist frequency.
Actually, Nyquist says that we have to sample faster than the signal bandwidth, not the highest frequency. But this leads us into multirate signal processing which is a more advanced subject.
