It's important to note that this is not a decomposition of $V$ as a representation of $SU(2)$ , only as a representation of $T$ . In particular, if you act with a nondiagonal matrix in $SU(2)$ then it will mix up these subspaces ${V}_{i}$ .
Weight space decomposition
Weight spaces
We now embark on our study of the fine structure of $SU(2)$ representations, aiming towards the classification theorem for irreducible representations that we stated earlier.
The first key observation is that $SU(2)$ contains a subgroup $T$ isomorphic to $U(1)$ : $$T=\{\left(\begin{array}{cc}\hfill {e}^{i\theta}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {e}^{i\theta}\hfill \end{array}\right):{e}^{i\theta}\in U(1)\}.$$
If I have a complex representation $R:SU(2)\to GL(V)$ then I can restrict it to $T$ and I get a representation ${R}_{T}:T\to GL(V)$ . By the classification of irreps of $U(1)$ , we find that $$V={V}_{1}\oplus \mathrm{\cdots}\oplus {V}_{n}$$ where $n=dimV$ and, with respect to this splitting, $R\left(\begin{array}{cc}\hfill {e}^{i\theta}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {e}^{i\theta}\hfill \end{array}\right)$ is the diagonal matrix $\left(\begin{array}{ccc}\hfill {e}^{i{m}_{1}\theta}\hfill & \hfill \hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \mathrm{\ddots}\hfill & \hfill \hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill {e}^{i{m}_{n}\theta}\hfill \end{array}\right)$ . The ${m}_{1},\mathrm{\dots},{m}_{n}$ are integers called the weights, the ${V}_{i}$ are called weight spaces, and the direct sum is called the weight space decomposition.
Example
Take the standard representation $R:SU(2)\to GL(2,\mathbf{C})$ . We have $R\left(\begin{array}{cc}\hfill {e}^{i\theta}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {e}^{i\theta}\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill {e}^{i\theta}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {e}^{i\theta}\hfill \end{array}\right)$ . This is already diagonal. The weight spaces are spanned by the vectors $(1,0)$ and $(0,1)$ and the weights are $1$ and $1$ respectively.
We will depict the weight space decomposition as follows (the weight diagram). Consider the integer points sitting on the number line. Colour in each integer $m$ which appears as a weight, and label it by the number of ${V}_{i}$ having weight $m$ if this number is strictly bigger than 1. In the example above, the weight diagram is a blob at $1$ and a blob at $1$ .
Consider the representation $R:SU(2)\to GL(\U0001d530\U0001d532(2)\otimes \mathbf{C})$ , $R(g){M}_{v}=g{M}_{v}{g}^{1}$ . We computed the associated Lie algebra homomorphism ${R}_{*}:\U0001d530\U0001d532(2)\to \U0001d524\U0001d529(3,\mathbf{C})$ . We found that $${R}_{*}\left(\begin{array}{cc}\hfill ix\hfill & \hfill y+iz\hfill \\ \hfill y+iz\hfill & \hfill ix\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 2z\hfill & \hfill 2y\hfill \\ \hfill 2z\hfill & \hfill 0\hfill & \hfill 2x\hfill \\ \hfill 2y\hfill & \hfill 2x\hfill & \hfill 0\hfill \end{array}\right).$$ Since we're interested in the diagonal subgroup $T$ we can set $y=z=0$ . We're interested in $$R\left(\mathrm{exp}\left(\begin{array}{cc}\hfill ix\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill ix\hfill \end{array}\right)\right)=\mathrm{exp}\left({R}_{*}\left(\begin{array}{cc}\hfill ix\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill ix\hfill \end{array}\right)\right)$$ which gives $$\mathrm{exp}\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2x\hfill \\ \hfill 0\hfill & \hfill 2x\hfill & \hfill 0\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mathrm{cos}2x\hfill & \hfill \mathrm{sin}2x\hfill \\ \hfill 0\hfill & \hfill \mathrm{sin}2x\hfill & \hfill \mathrm{cos}2x\hfill \end{array}\right).$$ The weights are going to be $2,0,2$ . This is because:

We calculated the weights of the representation $U(1)\to GL(2,\mathbf{C})$ , ${e}^{i\theta}\mapsto \left(\begin{array}{cc}\hfill \mathrm{cos}\theta \hfill & \hfill \mathrm{sin}\theta \hfill \\ \hfill \mathrm{sin}\theta \hfill & \hfill \mathrm{cos}\theta \hfill \end{array}\right)$ : these were $\pm 1$ because with respect to a basis of eigenvectors this matrix diagonalised and became $\left(\begin{array}{cc}\hfill {e}^{i\theta}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {e}^{i\theta}\hfill \end{array}\right)$ .

We're looking at the same submatrix but with $2\theta $ everywhere instead of $\theta $ , so this submatrix can be diagonalised to give $\left(\begin{array}{cc}\hfill {e}^{i2\theta}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {e}^{i2\theta}\hfill \end{array}\right)$ .

What's left is the topleft entry which is $1={e}^{i0\theta}$ , so we also get a weight space with weight 0.
The weight diagram is therefore:
Take ${\mathrm{Sym}}^{2}({\mathbf{C}}^{2})$ . If ${e}_{1}$ and ${e}_{2}$ form the standard basis for the standard representation then this has basis ${e}_{1}^{2}$ , ${e}_{1}{e}_{2}$ , ${e}_{2}^{2}$ . Since $\left(\begin{array}{cc}\hfill {e}^{i\theta}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {e}^{i\theta}\hfill \end{array}\right)$ acts as $${e}_{1}\mapsto {e}^{i\theta}{e}_{1}\text{and}{e}_{2}\mapsto {e}^{i\theta}{e}_{2}$$ in the standard representation, ${\mathrm{Sym}}^{2}\left(\begin{array}{cc}\hfill {e}^{i\theta}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {e}^{i\theta}\hfill \end{array}\right)$ acts as $${e}_{1}^{2}\mapsto {e}^{i2\theta}{e}_{1}^{2},{e}_{1}{e}_{2}\mapsto {e}_{1}{e}_{2},{e}_{2}^{2}\mapsto {e}^{i2\theta}{e}_{2}^{2}$$ (each factor transforms under the standard representation and then you multiply them together and the factors of ${e}^{i\theta}$ or ${e}^{i\theta}$ either combine or cancel).
The weight spaces are therefore:

${V}_{1}=\mathbf{C}\cdot {e}_{1}^{2}$ , with weight $2$ ,

${V}_{2}=\mathbf{C}\cdot {e}_{1}{e}_{2}$ , with weight $0$ ,

${V}_{3}=\mathbf{C}\cdot {e}_{2}^{2}$ , with weight $2$ .
The weight diagram is therefore the same as the weight diagram for the previous example:
We'll see later that this is enough to tell us that these representations are isomorphic.
Preclass exercise
Figure out the weight diagram for ${\mathrm{Sym}}^{n}({\mathbf{C}}^{2})$ .