Fibrant replacement is essentially sheavification with respect to the corresponding topology. So étale K-theory is nothing more than the part of K-theory that satisfies étale descent. Concretely (and computationally) this means that you have a spectral sequence (the descent spectral sequence)

$$ H^*(X_{ét},\pi_*K)\Rightarrow \pi_*K^{ét}(X) $$

(this is the analogue of Čech cohomology computations for sheaves of spectra) and similarly for the other topologies (except it turns out that algebraic K-theory is already a sheaf for the Nisnevich topology and not only for the Zariski topology: this is a consequence of the localization sequence).

One very cool fact, due to Thomason, is that étale sheavification for algebraic K-theory turns out to be the same thing as inverting a Bott element (at least after localizing away from the characteristic of the base field).

Whether this is "geometric" or not I guess it depends on whether you are used to think of sheaf theory as geometric or not.