The successive electrons in the experiment are assumed to be statistically independent; they need to be "unrelated" to each other. Physically, we try to do this by various methods, such as
- production from a hot (i.e. noisy) thermal system (consider that the classic "electron gun" is essentially an incandescent lightbulb plus extras)
- separation in time
- separation in space (mostly follows from time separation if the electrons are fast)
If the electrons are still related to/communicating with each other after we've done everything we can to separate them, then there's some force/phenomenon responsible for it which we should be able to detect. If we're not seeing anything then the electrons should be independent.
Once we have a supply of independent electrons, we can perform the double-slit experiment on them, taking care to keep them separate and to keep the apparatus itself from "remembering" anything between shots. Once we are done, we will be able to say "it was a property of our electron collection that it produced so-and-so distribution on our detector screen." Because our electrons were independent, we can go further and say "any one electron put through the experiment will have so-and-so probability of landing in any given position."
(The precise reasoning for the above is: for any one electron, there is some probability distribution for where it will land on the screen, even if we don't know it. If we have a beam of independent electrons, then the pattern it makes on the screen is completely determined by the single-electron distribution. If we measure the pattern, we can invert the relation to learn about the single-electron distribution. Without independence, the relation between beam pattern and single-electron distribution is broken.)
It is of course impossible to do the double-slit experiment with one electron and get a result of much confidence. Compare a similar macroscopic question:
You have a coin that will blow itself up after you flip it once. Can you tell whether the coin is fair (50/50) or not by flipping it?
The electron is consumed by the detector screen analogously to this coin. With just one trial, you cannot get a good picture of the probability distribution.