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Basically, what happened is that we managed to measure time very accurately with the advent of atomic clocks. Einstein's theory of relativity predicts, or rather posits, that the speed of light is an invariant. So measuring time accurately implies we can measure distances accurately too. We only have to check that light is indeed an invariant and there are also easily reproducible and precise lab techniques to check that, for instance by interferometry. This allows then to define the meter. The number 299,792,458 is then chosen so to make the meter agree with the old definition which used a standard rod.

2017105 — Basically, it is only showing what the eye will see from that specific spot in the visual space. If the beam of light were coming from a ...

You simply point the laser distance measuring devices dot at a target like a wall, a house, a utility pole, nearly any object, and the device will show the ...

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In a way,'speed of light' is akin to 'right angle': the concept itself is important but the associated value is void of real meaning as it is based on a rather arbitrary convention. A right angle is agreed (not measured!) to equate to 90 degrees, and similarly the speed of light is agreed (not measured!) to equate to 299792458 meters per second.

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I just complete @Chad Orzel's response by point : you seem to be worried on the effect of chosen frame of reference and space-time curvature. One of the reasons to redefine the meter in term of speed of light is because the speed of light is independent of the frame of reference and the space time curvature, while distance and time are not. That the whole point of Einstein relativity theories.

The particular values of the meter and the speed of light that we choose are based on older measurements using a meter defined in terms of the circumference of the Earth. We've chosen to keep that value, because it would be a hassle to make a wholesale change.

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If the metre is now defined as the distance light travels in vacuum in $1/299\,792\,458^{\textrm{th}}$ of a second and the speed of light is accepted to be $299\,792\,458\ \textrm{m}\,{\rm s}^{-1}$, doesn't this seem like the chicken and egg problem?

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The speed of light is invariable but the number or ratio of the frames of reference are variable as determined by the amount of gravity, the result of which is gravitational lensing.

There are three relevant quantities involved here: the length of a meter, the duration of one second, and the speed of light. You only need to absolutely measure one of them, after which the other two can be defined in terms of the one that is measured.

So how, then, do physicists claim to have accurately measured the speed of light? Does this mean the definition of the metre is dependent on our ability to accurately measure the speed of light?

What about so many other factors that affect these measurements? I do not claim to understand the theories of relativity entirely, but what about the chosen frame of reference? Spacetime curvature?

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For technological reasons, we have chosen to make the measured reference quantity the length of one second, which is defined in terms of the number of oscillations of radiation associated with the transition between the hyperfine ground states in cesium (specifically, it's 9,192,631,770 oscillations of that light). This is basically because there are experimental techniques that allow incredibly precise measurements of the frequency of radiation, at a level that really can't be matched by length or speed measurements. (The best frequency measurements in the world use trapped aluminum ions as the "clock," and are good to something like one part in $10^{18}$.)

The speed of light is not a fundamental constant and it does not represent a measured quantity. It is a conversion factor that creeps in due to the use of a clumsy system of units that uses different units for distance depending on direction (spatial vs temporal) in spacetime.

Sep 27, 2023 — Aberration is the effect of the transverse speed of the Earth added to the speed of light coming from a star. For instance, this effect on the ...

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It seems that in practice, the meter is often determined by measuring wavelengths of certain atomic transitions, read further on:

The agreed value of speed of light, combined with the definition of the second, results in an unambiguous unit of length. In other words, by defining the unit of speed, units for both length and time are defined based on a frequency standard. There is no circular reasoning involved in this.

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2021426 — In non-polarized light, for any particular photon/wave, the orientation is random. Some up-down, some side-to-side, and everything in between.

Having defined the second in terms of some physically measurable quantity, we are then free to define the speed of light as having some particular value in meters/second, and then define the meter in terms of the distance traveled by light in one second. The size of a meter is merely a matter of convention, not anything fixed in the physical world, so as long as we have anchored the second to something fundamental, we can make the meter be whatever we want.

After all, to determine speed (distance traveled/time taken) you must first choose some standards of distance and time, and so different choices can give different answers?