I am trying to prove with the differential definition that isometries preserve geodesics. That is: Let $c(t)$ be a geodesic in a $n$-dimensional semirriemannian manifold $M$ and $g$ an isometry of $M$ on itself. Then $g\circ c$ will be a geodesic if $$ \frac{d^2y^k}{dt^2}+\Gamma^k_{ij}\frac{dy^i}{dt}\frac{dy^j}{dt}=0,\ \ k=1,...,n,\ (\ast) $$ where $(\phi\circ g\circ c)(t)=(y^1(t),...,y^n(t)$ for a chart $\phi$ of $M$.
Doing some research online I found Isometries preserve geodesics . Here they talk about connections. I've looked for a definition (that allowed me to understand them) of connection for quite a while but so far the whole concept makes no sense to me.
Why is it necessary to use connections to prove $(\ast)$? Can $(\ast)$ be proved without using them? What is a connection? Is a concrete mathematical object like a set with a concrete structure or it is just a "meta-object" just to say that "we transport data along a curve"? I am really confused about the connection topic but I'm just a beginner in differential geometry so I guess I'm missing some obvious fact.
Thank u in advance. Any help would be appreciated.