What does it mean when a chern class of a vector bundle is postive(resp. negative)?

Recently i was studying line bundles on $$\mathbb{C}P^1$$. Here is my confusion: for any holomorphic map $$f:\mathbb{C}P^1 \to M$$, where $$(M,E,\nabla)$$ is a $$r$$-rank holomorphic vector bundle with a linear connection $$\nabla$$. Then $$f^*E$$ split into direct sums of $$r$$ line bundles $$L_1\oplus\cdots \oplus L_r$$ (Why??? any reference to this?)
If $$c_1(E)$$ is positive, since $$f^*c_1(E)=c_1(L_1)+\cdots+c_1(L_r)$$, then $$c_1(L_i)$$ is positive for some $$i$$.

So what does it mean a first chern class (as a cohomology class of $$H^2(M)$$) is positive? and why does a sum of first chern classes is positive indicates some of them must be positive? If there is any reference to this all, please tell me.

We say $$c_1(E)$$ is positive if, when regarded as an element of $$H^2_{\text{dR}}(X)$$, it can be represented by a Kähler form, i.e. a closed positive real $$(1, 1)$$-form. We say $$c_1(E)$$ is negative if $$-c_1(E)$$ is positive. We could also have $$c_1(E) = 0$$ in $$H^2_{\text{dR}}(X)$$, but in general $$c_1(E)$$ is neither positive, negative, nor zero.
In the case of $$\mathbb{CP}^1$$, the situation simplifies. First we have an isomorphism $$\Phi : H^2(\mathbb{CP}^1; \mathbb{Z}) \to \mathbb{Z}$$ given by the orientation on $$\mathbb{CP}^1$$ induced by the complex structure. Then $$c_1(E) \in H^2(\mathbb{CP}^1; \mathbb{Z})$$ is positive in the above sense if and only if $$\Phi(c_1(E)) > 0$$; similarly, $$c_1(E)$$ is negative if and only if $$\Phi(c_1(E)) < 0$$. In particular, every holomorphic line bundle over $$\mathbb{CP}^1$$ is of the form $$\mathcal{O}(a)$$ for some $$a \in \mathbb{Z}$$ and $$\Phi(c_1(\mathcal{O}(a))) = a$$, so $$\mathcal{O}(a)$$ is positive if and only if $$a > 0$$. By the Grothendieck lemma, every holomorphic vector bundle $$V \to \mathbb{CP}^1$$ splits as a sum of line bundles $$V \cong \mathcal{O}(a_1)\oplus\dots\oplus\mathcal{O}(a_r)$$. Then
\begin{align*} \Phi(c_1(V)) &= \Phi(c_1(\mathcal{O}(a_1)\oplus\dots\oplus\mathcal{O}(a_r)))\\ &= \Phi(c_1(\mathcal{O}(a_1)) + \dots + c_1(\mathcal{O}(a_r)))\\ &= \Phi(c_1(\mathcal{O}(a_1))) + \dots + \Phi(c_1(\mathcal{O}(a_r)))\\ &= a_1 + \dots + a_r. \end{align*}
So $$c_1(V)$$ is positive if and only if $$a_1 + \dots + a_r > 0$$; note, this implies that $$a_i > 0$$ for some $$i$$, and hence $$c_1(\mathcal{O}(a_i))$$ is positive. Now apply the above to the bundle $$V = f^*E$$ (the line bundles $$L_j$$ you mention are the $$\mathcal{O}(a_j)$$).