Sections of invertible sheaf (for tautological line bundle) on $\Bbb{CP}^1$ Let $E$ be the total space for the tautological line bundle $p:E\to \Bbb{CP}^1$.
I see that $p^{-1}(x)=\{x\}\times \{v\}_{v\in x}\cong \{x\}\times\Bbb A^1$ (where $x\in \Bbb P^1$ and I denoted $x$ also as a $1$-dimensional subspace of $\Bbb A^2$, living in $G(1,\Bbb A^2)$. So that I would expect a section to be a map $s$ taking $s:x\mapsto \{x\}\times v$ where $v\in \Bbb A^1$, $s:\Bbb{CP}^1\to \Bbb A^1$.

My book says that the invertible sheaf $L_{\Bbb{CP}^1}(E)$($=L(E)$ in future) assigns for each open $U\subset \Bbb{CP}^1$ the collection of regular functions on $p^{-1}(U)$ which are linear on fibres of $p$.

This doesn't make sense to me. When we talk about sections for a sheaf, we mean $s\in L(U)$, and when we talk for a vectorbundle, we mean $s:\Bbb{CP}^1\to E$.
Here they are saying my sections are maps $s:p^{-1}(U)\to \Bbb A^1$, I would assume, which doesn't make sense in either interpretation of sections.
How does the shaded part above (from my book) make sense?
 A: Let me try my best to explain how I think about $E$ and $L(E)$.
Let us use coordinates $[X:Y]$ for $\mathbb {CP}^1$. We will cover $\mathbb {CP}^1$ with two affine sets, $U_0 = \{[1:z] : z \in \mathbb C \}$ and $U_1 = \{ [w:1] : w \in \mathbb C \}$, so $z = 1/w$ on the overlap $U_0 \cap U_1$.
Now I'm going to describe the line bundles $E$ and $L(E)$ by specifying the transition functions on $U_0 \cap U_1$ in both cases. (Specifying transition functions is always enough to fully define a line bundle.)
First, $E = \{ ([X:Y],(x,y)) \in \mathbb {CP}^1 \times \mathbb C^2 : xY = yX \}$. To trivialise this, let us parametrise $E$ by writing $([X,Y],(x,y)) = ([1:z],(u,uz))$ on $U_0$ and $([X,Y],(x,y)) = ([w:1],(vw,v))$. Thus $u $ is the fibre coordinate on $U_0$ and $v$ is the fibre coordinate on $U_1$. These fibre coordinates are related by $v = uz = u/w$ on the overlap $U_0 \cap U_1$. In other words, the transition function for $E$ with respect to my chosen trivialisation is $z = 1/w$.
Next, we define $L(E)$ to be the dual of the line bundle $E$. Since $L(E)$ is the dual of $E$, its transition function with respect to the same trivialisation must be the inverse of the transition function for $E$, i.e. the transition function on $U_0 \cap U_1$ is $1/z = w$. This is enough to define $L(E)$.
But let us check this with a specific example anyway, in order to help you connect with your book. Saying $L(E)$ is the dual of $E$ is the same as saying that a global section of $L(E)$ is a collection of linear functionals on the fibres of $E$, one for each fibre, chosen so that these linear functionals vary from one fibre to the next in a holomorphic way. Consider the global section $s$ of $L(E)$, which acts linearly on each fibre of $E$ by sending each $(x,y)$ to $ax + by$, where $a,b$ are fixed constant complex numbers. On the $U_0$ patch, $s$ sends $u \mapsto (a+bz)u$, i.e. $s$ is represented by the $1\times 1$ matrix $a+bz$. On the $U_1$ patch, $s$ sends $v \mapsto (aw+b)v$, i.e. $s$ is represented by the $1\times 1$ matrix $aw+b$. Indeed, $a+bz$ and $aw+b$ are both holomorphic, which shows that $s$ is a legitimate holomorphic section of $L(E)$. In this example, you can think of $a+bz$ and $aw+b$ as the trivialised forms of $s$ over the two patches. The transition function between the two patches is the ratio between these two trivialised forms, which is $1/z = w$, as claimed earlier.
It is perfectly possible too to write down sections of $L(E)$ that are defined on local patches of $\mathbb{CP}^1$ but not globally. For example, $(x,y) \mapsto (ax^2+bxy+cy^2)/x$ is a section of $L(E)$ defined on $U_0$, but has a pole at $[0:1]$; its trivialised forms are $a + bz + cz^2$ over $U_0$ and $aw + b + c/w$ over $U_1$ and the pole occurs at $w = 0$.
By the way, everybody refers to $E$ and $L(E)$ as $\mathcal O(-1)$ and $\mathcal O(1)$ respectively. I hope this helps.
