The connection is via the additivity theorem in (connective) algebraic $K$-theory or, better, the fact that algebraic $K$-theory is the universal additive invariant as proven in BGT. Let me try to give an answer that blackboxes $K$-theory in terms of its universal property.
So Beilinson provided a semiorthogonal decomposition of the derived category of coherent sheaves $\mathbb{P}^n$ which reads as $D(Coh(\mathbb{P}^n)) \simeq D(Coh(k)) \oplus ... D(Coh(k))$. Just as decompositions of an abelian group comes from splitting of short exact sequences, this decomposition takes place as the splitting of "short exact sequences" in some category. One form of this category is the category of idempotent complete small stable $\infty$-categories which 1 denotes by $Cat^{perf}_{\infty}$ and a short exact sequence is really a cofiber sequence in this category which splits (morphisms are exact functors, a cofiber sequence $A \rightarrow B \rightarrow C$ roughly witnesses $C$ as a Verdier quotient of $B$ by $A$ up to idempotent completion, splitting means that there appropriate adjoints splitting the sequence above). If you are more inclined with the $dg$-language, this is also explained nicely in Tabuada's survey.
Anyway algebraic $K$-theory is a functor $Cat^{perf}_{\infty} \rightarrow Spectra$. By the additivity theorem (see also [BGT, Proposition 7.10]) this functor carries split cofiber sequences to sums. It turns out that it is actually the "universal" such functor (see 1 for precise statements). So the Beilinson decomposition above is yet another way to see that $K(\mathbb{P}^n_k) \simeq K(k) \oplus ... K(k)$, ditto for the quadrics and Grassmanian stuff
To summarize: if you find a nice set of generators for your derived category, then you can hope for a semiorthogonal decomposition. This is interpreted as a split cofiber sequence in some category and algebraic $K$-theory eats in stuff in this category and takes these cofiber sequences to sums in spectra. Furthermore this property "characterizes" the $K$-theory functor.
