Maybe the best way to think of it is that a particle, like an electron, can only be viewed as an actual particle as an approximation to reality in certain circumstances. It maybe better to think of it as a fuzzy ball with its fuzziness extending to infinity. This fuzziness being a measure of the probability of its actual position.

When you send an electron towards a pair of slits, there is no meaning in trying to determine "which it will go through". The whole fuzzy caboodle is heading in a very rough sort of direction. The result on the other side will be that there will be a pattern of probable positions where it is most likely that the electron will end up. It so happens that the probability function of a moving electron is characterised by a wave so that in passing through two slits we see this probability wave interfering and producing an array of probability "lines" on the far side of the slits. This result can be calculated easily in this case.

For a more complex system, with numerous possible paths for the electron to have a probability to pass along, the calculations become a little harder because each path has a different probability and the way the waves interfere can be more difficult to calculate. Nonetheless it is possible. Because you generally know that your electron will probably (hugely probably) be simply be considered to move along the paths you have defined, the calculation is finite. In fact ALL the paths for the electron (because the electron is a fuzzy ball of probability) are infinite in number even if most have negligible probability for practical measurements. So even with no apparatus, when an electron is moving from one place to another, the probability of its position is spread throughout space.

To calculate the probability of an electron being detected at a particular position is therefore to take the sum of all the probabilities which will encompass all paths throughout space. When the electron has been detected you know the sum of these probabilities is equal to one, because it has happened. In this sense you could say that the electron has taken all possible paths. This is when the wavefunction "collapses" because it is a wave of probability that no longer exists when the outcome is known.

A lot (if far from all) becomes clearer if you think that the probability functions are there whether the electron is there or not, rather like the probability of tossing a coin and turning up a head (or a tail) is a half. You can calculate what would "probably" happen in a certain circumstance as a fuction of the apparatus (or structure of space). When the event takes place this has resolved to a certain outcome and it is not so surprising that this is throughout space.