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The dimension of $\mathbb{R}$ as a vector space over $\mathbb{Q}$ is equal to the cardinality of $\mathbb{R}$.
In general, the dimension and cardinality of any vector space $\mathbf{V}$ and the cardinality of its scalar field $\mathbf{K}$ will obey the following equation:
$$
\|\mathbf{V}\| = \left\\{
~~\begin{array}{ll}~~
\|\mathbf{K}\|^{\dim(\mathbf{V})} & \text{$\dim(\mathbf{V})$ is finite} \\\\
\max(\|\mathbf{K}\|, \dim(\mathbf{V})) & \text{$\dim(\mathbf{V})$ is infinite}
\end{array}
~~\right.~~
$$
In the case of $\mathbb{R}$ over $\mathbb{Q}$, if the dimension of $\mathbb{R}$ were somehow finite then this equation would imply that $|\mathbb{R}|$ is equal to some finite power of $|\mathbb{Q}|$, which is false (every finite power of a countable set is countable). Since $|\mathbb{R}|$ is greater than $|\mathbb{Q}|$, the second branch of the equation implies that the dimension of $\mathbb{R}$ is equal to $|\mathbb{R}|$.

The dimension of $\mathbb{R}$ as a vector space over $\mathbb{Q}$ is equal to the cardinality of $\mathbb{R}$.
In general, the dimension and cardinality of any vector space $\mathbf{V}$ and the cardinality of its scalar field $\mathbf{K}$ will obey the following equation:
$$
\|\mathbf{V}\| = \begin{cases}
\|\mathbf{K}\|^{\dim(\mathbf{V})} & \text{$\dim(\mathbf{V})$ is finite} \\\\
\max(\|\mathbf{K}\|, \dim(\mathbf{V})) & \text{$\dim(\mathbf{V})$ is infinite}
\end{cases}$$
In the case of $\mathbb{R}$ over $\mathbb{Q}$, if the dimension of $\mathbb{R}$ were somehow finite then this equation would imply that $|\mathbb{R}|$ is equal to some finite power of $|\mathbb{Q}|$, which is false (every finite power of a countable set is countable). Since $|\mathbb{R}|$ is greater than $|\mathbb{Q}|$, the second branch of the equation implies that the dimension of $\mathbb{R}$ is equal to $|\mathbb{R}|$.