The below is from the reference but there is no proof. Is it true? Why?
Claim: For sphere or an ellipsoid $(K > \delta > 0)$ geodesics which start from the same point, re-approach each other and become tangent to the conjugate locus $C(p)$.
Reference: Differential Geometry of Curves and Surfaces Manfredo P. do carmo
For the surface $S$ let $K$ be the Gaussian curvature of $S$ and conjugate means the Jacobi conjugate.
Definition: Let $y: [0, I] \to S$ be a geodesic of $S$ with $\gamma(0) = p$. We say that the point $q = \gamma(s_0)$, $s_0 \in [0, 1]$, is conjugate to $p$ relative to the geodesic $\gamma$ if there exists a Jacobi field $J(s)$ which is not identically zero along $y$ with $J(0) = J(s_0) = 0$.
In general, given a point $p$ of a surface $S$, the "first" conjugate point $q$ to $p$ varies as we change the direction of the geodesic passing through $p$ and describes a parametrized curve. The trace of such a curve is called the conjugate locus to $p$ and is denoted by $C(p)$.
The snippet you quote is merely a definition of $C(p)$. If you'd like to see what $C(p)$ looks like for an ellipsoid, see Fig. 19 on the Wikipedia article on Geodesics on an ellipsoid. There are a couple of possible answers to why initially neighboring geodesics intersect: (1) for $K > 0$, the surface is closed so that they can't avoid each other indefinitely; (2) the solution to Gauss's equation for the separate $d^2m/ds^2 + Km = 0$ is oscillatory for $K > 0$.