Why $\{\mathbf{0}\}$ has dimension zero?

A vector by itself doesn't have a dimension. A subspace has a dimension. Why $\{\mathbf{0}\}$ is considered as having dimension $0$? Because of consistency with all other situations. For instance $\mathbb{R}^3$ has dimension $3$ because we can find in it a linearly independent set with three elements, but no larger linearly independent set. This applies to vector spaces having a finite spanning set and so of subspaces thereof.

What's the largest linearly independent set in $\{\mathbf{0}\}$? The only subsets in it are the empty set and the whole set. But any set containing the zero vector is linearly dependent; conversely, the empty set is certainly linearly independent (because you can't find a zero linear combination with non zero coefficients out of its elements). So the only linearly independent set in $\{\mathbf{0}\}$ is the empty set that has zero elements.

The sum over the empty set is the additive identity... in this case zero.

If $~0~$ is a part of your basis, and if it is the only vector, there are zero linearly independent vectors in it, since any non-negative multiple of $~0~$ is $~0~$.

Thus, the dimension of the space is $~0~$.