Why are coherent sheaves on $\Bbb P^1$ derived equivalent to representations of the Kronecker quiver?

Let $\mathcal O$ be the structure sheaf of $\mathbb P^1$. Then $\mathcal O \oplus \mathcal O(1)$ is rigid and generates the derived category of coherent sheaves on $\mathbb P^1$. Thus, it is a tilting object, and so the derived category is equivalent to the category of modules over its endomorphism ring, which is the path algebra of the Kronecker quiver.

For $\mathbb P^n$, you need $n+1$ objects to generate the derived category. You can take $\mathcal O(i)$ for $0\leq i \leq n$; this will be a tilting object again, but its endomorphism ring will not be hereditary; you will get a quiver with relations. This shouldn't be surprising, though: path algebras of quivers have global dimension one, so you shouldn't expect their derived categories to agree with derived categories of sheaves on higher-dimensional varieties.

The original reference for the general result is

A.A. Beilinson, Coherent sheaves on $P^n$ and problems of linear algebra, Func. Anal. Appl. 12 (1978), pp. 214-216.

Google search brings many related papers and lecture notes, e.g. a very elementary exposition http://math.harvard.edu/~hirolee/pdfs/280x-13-14-dbcoh.pdf , and also the paper https://arxiv.org/pdf/math/0702861.pdf where in the very beginning you see the quiver with relations needed for $P^n$ (your guess, as you will see in that paper, is not quite correct).

One version of the question you might be asking is where such equivalences come from in general, as opposed to a proof of this one in particular. The general setup is "derived Morita theory": here is an underived version of the basic result.

As setup, let $C$ be a cocomplete abelian category and let $S$ be an essentially small full subcategory of $C$. This gives a restricted Yoneda embedding

$$Y : C \ni c \mapsto (s \mapsto \text{Hom}(s, c)) \in [S^{op}, \text{Ab}]$$

and we would like to know when this is an equivalence of categories. Roughly speaking this means that $S$ "freely generates" $C$.

Theorem (Freyd): The restricted Yoneda embedding is an equivalence of categories if and only if the following conditions hold:

• each object $s \in S$ is compact projective, and
• if $Y(c) = 0$, then $c = 0$.

For a long meandering proof see this blog post. These conditions are unfortunately quite restrictive: for example, for modules compactness is equivalent to being finitely presented, so the only compact projectives are the finitely presented projectives. Note that

• if $S$ has one object $s$ the conclusion of the theorem is that $C$ is equivalent to the category of right $\text{End}(s)$-modules,
• if $S$ has finitely many objects $s_1, \dots s_n$ a small argument shows that the conclusion of the theorem is that $C$ is equivalent to the category of right $\text{End}(s_1 \oplus \dots \oplus s_n)$-modules, and
• if $S$ is the free $\text{Ab}$-enriched category on a quiver the conclusion of the theorem is that $C$ is equivalent to the category of (right) representations of this quiver (over $\mathbb{Z}$, but you can replace $\text{Ab}$ with vector spaces and everything goes through the same).

Applications of this theorem include Serre's affineness criterion and, I believe, the Dold-Kan theorem, although I haven't seen the details written down.

Now, there is an analogous theorem in the derived (stable $\infty$) setting which has the pleasant additional property that the projectivity hypothesis can be dropped. This makes it much easier to find full subcategories $S$ satisfying the hypotheses of the theorem; see, for example, Schwede-Shipley for a discussion in the setting of model categories. In particular, many schemes (stacks, etc.) now have the property that their derived categories of quasicoherent sheaves satisfy the hypotheses of the theorem; these are said to be "compactly generated."