When calculating propulsive manoeuvres, we usually consider them as impulses. That is, happening in a very short time. This is usually a good enough model for chemical rocket engines which deplete their propellant in minutes or even seconds.
So, usually: Place the burn started $\approx$ place the burn ended.
This can be combined with a fundamental property of orbits: They repeat.
So exactly one orbit later, we are back in the original location, regardless of what propulsive impulse was performed there.
If, on the other hand, your burn takes a while, like with an ion engine or with a small motor needing to burn through a lot of propellant, the original location does as a matter of fact not stay unaffected.
For the apoapsis raising manoeuvre specifically
You are at the periapsis, so your orbit has only tangential velocity and no radial velocity. By adding more, you still only have tangential velocity.
There are only two points in an elliptic orbit with tangential velocity only, the periapsis and the apoapsis. So if what we are raising is the apoapsis, the current point must by necessity be the periapsis, and thus remain unchanged. (you could flip them though, by doing a retrograde burn to take the tangential velocity below the circular orbital velocity, making the current point the apoapsis). By angling the burn a little, so it also adds radial velocity, would indeed be changing the periapsis location.