How does the electron understand that it being observed in the double slit experiment?

One way to "observe" a tiny particle like an electron is by detecting its presence via its electric field. That detection necessarily requires that the electron disturbs some part of the the detection device's electric field if it is to be registered by that detection device. Due to Newton's third law, the electron must be similarly disturbed. Or if you prefer to "see" that electron with a photon, you must necessarily use a photon with a very short wavelength (aka very high energy) because the electron is so small. That high energy photon will also disturb the electron when it reflects off of it, due to conservation of momentum. In other words, the electron does not "understand" that it is being observed ... it is so very tiny that any force that interacts with it such that you can determine its position, will change its behavior, unlike common macroscopic objects which are so very massive that bouncing photons off of them has no discernible effect.

Electrons are quantum mechanical entities, and interact quantum mechanically with the environment. This means that there are differential wave equations, whose solutions control the probability of how an electron will interact . Probability means that an accumulation of the same boundary condition events should be made, in order tos see an effect.

In the double slit case , the boundary conditions are "electron falling on double slit given distances separation and width of slits". This is controls the boundary conditions that choose the particular wave-function solution. It becomes a different experiment if the electron is disturbed in order to detect which slit it went through, different boundary conditions and thus different wavefunctions .

In different words, in order to detect an electron, an interaction has to happen, all interactions disturb the original boundary conditions, and a different wavfunction will control the track of the electron destroying the coherence needed in order to sum many electrons with the same boundary conditions.

We do not know how does it know, we just experimentaly proven that it knows. The experiment is delayed-choice quantum eraser experiment. It basicaly does double slit experiment with a trick to produce two photons out of one AFTER it gone through the slits.

One photon travels to the detector where we observe or do not observe interference. The other photon travels to an array of detectors in which it has 50% chance to go to a detector that tells us which slit photon gone through, and 50% chance to go to a detector which erases that information and we no longer know which slit photon went.

Surpise surprise when we knew which slit photono came through there were no interferance paterns, when we didnt interferance paterns emerged.

Fun fact, the photons that told us which slit original photon came through hit sensors 8ns later than the one which generated paters. So the photon knew what we would know (or what we will not know(which slit it came through)) 8ns later. How? We do not quite know.

Here is video explaining experiment: https://www.youtube.com/watch?v=8ORLN_KwAgs