Geometric meaning of the Euler sequence on $\mathbb{P}^n$ (Example 8.20.1 in Ch II of Hartshorne)

Yes! The geometric picture is very nice and very easy. It is explained on pages 408-409 of Griffiths-Harris.

Here is roughly how it works:

Let's work over $\mathbb{C}$ for simplicity. Think of $\mathbb{P}^n$ as being the quotient of $X := \mathbb{C}^{n+1} - 0$ by the action of $\mathbb{C}^\ast$. On $X$ we have the vector fields $d/dx_i$, where the $x_i$ are the standard coordinates on $\mathbb{C}^{n+1}$. Check that if $v_i$ are linear functionals on $\mathbb{C}^{n+1}$, then the vector field $\sum v_i d/dx_i$ on $X$ descends to a vector field on $\mathbb{P}^n$. The surjection $\mathcal{O}(1)^{n+1} \to \mathcal{T}$ corresponds to taking $n+1$ linear functionals $v_i$ and projecting the vector field $\sum v_i d/dx_i$ down to $\mathbb{P}^n$. The kernel $\mathcal{O}$ corresponds to the vector field $E = \sum_i x_i d/dx_i$. Intuitively, $E$ is a "radial" vector field on $X$, and if you pretend that $\mathbb{P}^{n}$ is a "sphere" in $X$, then $E$ is "normal" to this "sphere", so it vanishes when we project it down.

Jonathan Wise gives a nice (and rigorous) explanation of this below.

Aside: I think the reason why this is called the Euler sequence is because the vector field $E$ is known as the Euler vector field. And perhaps the reason why $E$ is called the Euler vector field is because its flow is exponential, and $e = 2.718\dots$ is also known as Euler's number. But I'm not sure, and someone should correct me if I'm wrong about this. Edit: Today somebody told me that the relation $E f = d f$ for $f$ a homogeneous degree $d$ polynomial (as in Charles' answer) was discovered by Euler and is known as "Euler's relation".

To me, at least, this is a manifestation of the fact that for homogeneous polynomials, we have $\frac{1}{d}\sum_{i=0}^n x_i\partial_i f=f$. The map $O(1)^{n+1}\to T$ tells you that every vector field is a linear combination of the $\partial_i$ with linear coefficients. The map $O\to O(1)^{n+1}$ sends the section 1 to the vector $(x_0,\ldots,x_n)$, which says that the vector field $\frac{1}{d}\sum x_i\partial_i$ acts trivially on functions. Thus, the quotient must be the actual tangent vector fields, giving us the tangent bundle.

EDIT: Apparently this package doesn't know mathscr. Have made it readable.

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.