Let $(M,\nabla)$ be an affine manifold and $p\in M$. The exponential map at $p$ is defined as $\text{exp}_p:\mathcal{E}\to M$ such that $v\to \gamma_v(1)$ with $\mathcal{E}=\{v\in T_pM:\gamma_v(1) \text{ is defined }\}$ and $\gamma_v$ the unique geodesic s.t. $\gamma_v(0)=p$ and $\gamma_v'(0)=v$
In which book can I find a geometric motivation of this definition??