Two vector spaces $V$ and $W$ are said to be isomorphic if there exists an invertible linear transformation (aka an isomorphism) $T$ from $V$ to $W$.
The idea of a homomorphism is a transformation of an algebaric structure (e.g. a vector space) that preserves its algebraic properties. So an homomorphism of a vector space should preserve the basic algebraic properties of the vector space, in the following sense:
$1$. Scalar multiplication and vector addition in $V$ is carried over to scalar multiplication and vector addition in $W$:
For any vectors $x,y$ in $V$ and scalars $a,b$ from the underlying field, $T(ax+by)=aT(x)+bT(y)$.
$2$. The identity element of $V$ is carried over to the identity element of $W$:
If $0_V$ is the identity vector in $V$, then $T(0_V)$ is the identity vector in $W$.
$3$. Vector inversion in $V$ is carried over to vector inversion.
$T(-v)=-T(v)$ for all $v$ in $V$.
$1$ is precisely the property that defines linear transformations, and $2$ and $3$ are redundant (they follow from $1$). So linear transformations are the homomorphisms of vector spaces.
An isomorphism is a homomorphism that can be reversed; that is, an invertible homomorphism. So a vector space isomorphism is an invertible linear transformation. The idea of an invertible transformation is that it transforms spaces of a particular "size" into spaces of the same "size." Since dimension is the analogue for the "size" of a vector space, an isomorphism must preserve the dimension of the vector space.
So this is the idea of the (finite-dimensional) vector space isomorphism: a linear (i.e. structure-preserving) dimension-preserving (i.e. size-preserving, invertible) transformation.
Because isomorphic vector spaces are the same size and have the same algebraic properties, mathematicians think of them as "the same, for all intents and purposes."