what is the geometric significance of covariant differentiation?

Question

I studied tensor calculus and there is a chapter named covariant differentiation of tensors. There one defines the covariant differentiation of a rank $1$ contravariant tensor as $$A^i_{,j}=\frac{\partial A^i}{\partial x^j}+\Gamma^i_{kj}A^k$$ and that of a rank $1$ covariant tensor as $${A_i}_{,j}=\frac{\partial A_i}{\partial x^j}-\Gamma^k_{ij}A_k$$ and that of a rank $2$ mixed one as $$A^i_{j,k}=\frac{\partial A^i_j}{\partial x^k}+\Gamma^i_{mk}A^m_j-\Gamma^m_{jk}A^i_m$$ My question is, why we need those extra "+" and "-" adjustments accompanied by those second kind Christoffel symbols? We could have just written the first partial differentiation term as we did in vector analysis differentiation operators, like gradient, divergence, curl etc. In short, what do these Christoffel symbols add to the differentiation terms geometrically? Any help is appreciated.