Definition. The dimension of the configuration space is called the number of degrees of freedom.
Thus, if we find the dimension of the configuration space of a rigid body, we can deduce its degrees of freedom.
Definition. A rigid body is a system of point masses, constrained by holonomic relations expressed by the fact that the distance between points is constant:
$$|\mathbf{x}_i - \mathbf{x}_j| = r_{ij} = \text{const.}$$
Theorem. The configuration manifold of a rigid body is a six-dimensional manifold, namely, $\mathbb{R}^3 \times \operatorname{SO}(3)$ (the direct product of a three-dimensional space $\mathbb{R}^3$ and the group $\operatorname{SO}(3)$ of its rotations), as long as there are three points in the body not in a straight line.
The dimension of $\mathbb{R}^3 \times \operatorname{SO}(3)$ is indeed $3+3=6$.
Proof. Let $\mathbf{x}_1$, $\mathbf{x}_2$, and $\mathbf{x}_3$ be three points of the body which do not lie in a straight line. Consider the right-handed orthonormal frame whose first vector is in the direction of $\mathbf{x}_2-\mathbf{x}_1$, and whose second is on the $\mathbf{x}_3$ side in the $\mathbf{x}_1 \mathbf{x}_2 \mathbf{x}_3$-plane (Figure). It follows from the conditions $|\mathbf{x}_i - \mathbf{x}_j|=r_{ij}$ ($i=1,2,3$), that the positions of all the points of the body are uniquely determined by the positions of $\mathbf{x}_1$, $\mathbf{x}_2$, and $\mathbf{x}_3$, which are given by the position of the frame. Finally, the space of frames in $\mathbb{R}^3$ is $\mathbb{R}^3 \times \operatorname{SO}(3)$, since every frame is obtained from a fixed one by a rotation and a translation. $\blacksquare$
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Not only did we find the dimension of the configuration space, but exactly which space it is!
Intuition. You need 3 parameters from $\mathbb{R}^3$ in order to describe the position of the object (i.e. a point $\mathbf{x}$ in $\mathbb{R}^3$ which locates the position of the frame), and 3 parameters from $\operatorname{SO}(3)$ to describe its orientation (i.e. an element $R$ of $\operatorname{SO}(3)$ which defines the orientation of the frame). Thus, 6 parameters in total.
Mathematically, we can write that the configuration space (manifold) of a rigid body is the space defined by
$$\mathbb{R}^3 \times \operatorname{SO}(3) = \{(\mathbf{x},R):\mathbf{x} \in \mathbb{R}^3 \text{ and } R \in \operatorname{SO}(3)\}.$$
If you have trouble understanding the last bit of the proof, look at this question of mine. And if you are wondering why $\operatorname{SO}(3)$ is 3-dimensional, i.e. has 3 parameters, consider looking at this.
I hope that helps!
Source. Mathematical Methods of Classical Mechanics by V.I. Arnold.
