Elements higher in the activity series are more reactive; physically, what this means is that the element is more likely to want to give up its valence electrons. In a salt solution, e.g. AgCl, what is actually present in the solution is Ag+ and Cl- ions: the silver has given its one valence electron to chlorine. The element silver has a measurable tendency, or 'desire' to want to do this: this is its reactivity (silver actually has quite a low reactivity: it doesn't mind keeping its valence electron, whereas a element with high reactivity wants to get rid of its valence electron). If we introduce another metal, a more reactive one such as sodium, what we now have in the system is Ag+ and Na (forget the Cl- for the moment). As the sodium is more reactive, it has a greater desire to get rid of its valence electron than silver. There are two possibilities that could happen:
1. Nothing happens: the silver keeps its 1+ charge and sodium keeps its one valence electron which it desperately wants to get rid of, or
2. The sodium gives its valence electron to the silver, which doesn't mind having a valence electron (due to its low reactivity).
Which possibility do you think will happen? In possibility 1, silver is fairly stable; it doesn't really mind whether it has a 1+ charge (or a 0 charge), whereas sodium is very unstable and desperately wants to rid itself of its valence electron. Whereas in possibility 2, both elements are stable: the sodium has given up its valence electron and is thus stable, and silver is stable; it doesn't really mind what charge it has. Hence possibility two will happen, and sodium will 'displace' silver from the solution.
Hopefully that explanation helps. Excuse my extensive personification of elements; obviously silver and sodium aren't sentient beings, but it helps to pretend that they are.