Maybe your question is about the instrumentation used, in which case I cannot help much, but if it is about the concept of observing something in such experiments, then here's a short/rough discussion.
The observer is a black-box type of description of a more involved physical concept behind, namely that of interactions and measurements.
I guess you already know that we cannot visually see say an electron as that would imply probing the electron with the visible part of the light spectrum and in this range the wavelengths are too large compared to the system size to be probed.
But more generally, when we observe something experimentally, some form of interaction takes place, for example a scattering between the electron and a photon coming from a detector. In doing so we inevitably change the state of our system, and this is hidden behind that experimental act of observation.
To give you more intuition, take a particle under a microscope. The precision of position measurements of the particle is limited by the resolution of our microscope which itself is limited by the wavelength $\lambda$ of the light used. In order to improve this, $\lambda$ has to be made as small as possible, which results in using more energetic photons. The scattering that results between the photon and our system will jolt the system, and cause its momentum to change. All this process can be labeled as observing the position of a particle.
If you are interested in the instrumentation part, start by reading up about CCD's.
A neat explanation by Mark Eichenlaub can also be found on this Quora post, here's a small quote from it:
Let's give one example. In a real double-slit experiment, the
electrons are bounced off layers of atoms in a crystal, not slits.
Mostly, the electrons won't interact with the crystal except to bounce
off. The state of the crystal is unchanged. However, sometimes an
electron will interact with the crystal when it bounces, flipping the
spin of different electron in the crystal as it does so. In that case,
if we go to the crystal and see where the spin has been flipped, it
tells us where the electron bounced off. That means the spin-flipping
constitutes an observation of the electron, and electrons that flip a
spin in the crystal will not show interference. As long as there is
any system in the universe we can examine that keeps a record of where
the electron bounced, we destroy interference because the states don't
cancel any more.