Read about Riesz's lemma and its proof as given in the Functional Analysis text (2nd ed) by Taylor.

In the proof it's said that $X$ is the normed linear space. $X_0$ is its closed and proper subspace.

$x_1\in X-X_0$.

$x_0,x\in X_0$.

$h=||x_1-x_0||^{-1}$.

Then it's said that $$h^{-1}x+x_0\in X_0$$ How?

Since $h^{-1}$ is just some scalar, $h^{-1}x+x_0$ is just a linear combination of $x$ and $x_0$. Since $x,x_0\in X_0$ and $X_0$ is a linear subspace, this means $h^{-1}x+x_0\in X_0$.