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.
