Let us consider a vector space $E$ of dimension $n+1$ and a line $L\subset E$, which corresponds to the point 
$x\in \mathbb {P(E)}$.
The tangent space $T_x\mathbb P (E)$  is canonically isomorphic to  the space of linear maps $\mathcal{L}(L,E/L)$ 
[Harris,Algebraic Geometry, page 200, where it is even done  for Grassmannians].

Hence we get a canonical isomorphism $T_x\mathbb P (E)=L^\ast  \otimes E/L$ , transformed into
$T_x \mathbb P(E)\otimes L=E/L$

 
 which we write as  an exact sequence $0 \to L \to E \to T_x \mathbb P E \otimes L \to 0$

This was just over the point $x\in\mathbb P(E)$. If we globalize this over the whole of $\mathbb P(E)$ we get the exact sequence of locally free sheaves [Recall that the fibre at x of the tautological line bundle $\mathcal O (-1)$   is precisely $L$]

$0\to \mathcal O(-1) \to \mathcal O ^{n+1} \to T \mathbb P (E)\otimes \mathcal O (-1) \to 0$

By tensoring this exact sequence by the invertible sheaf $\mathcal O (1)$  we obtain the euler sequence.