The following is from Milnor's Topology from the Differentiable Viewpoint.
Problem 14. If $y\neq z$ are regular values for a [smooth] map $f:S^{2p-1}\to S^p$, then the manifolds $f^{-1}(y)$, $f^{-1}(z)$ can be oriented (they inherit an orientation from the one given to $S^p$), hence the linking number is defined. Prove that the linking number $l(f^{-1}(y), f^{-1}(z))$ is a locally constant function of $y$. The problem contains more parts after this to show that this linking number depends only on the homotopy class of $f$.
The linking number of two manifolds $M^m,N^n\subset S^{m+n+1}$ is obtained as follows. Take $p\in S^{m+n+1}\setminus(M\cup N)$ and identify $S^{m+n+1}-p$ with $\mathbb{R}^{m+n+1}$. Consider the linking map $\lambda:M\times N\to S^{m+n}$ given by $$ \lambda(x,y)=\frac{x-y}{||x-y||}, $$ where the difference and norm are taken as elements of $\mathbb{R}^{m+n+1}$. The linking number $l(M,N)$ is defined as the degree of this map $\lambda$.
If anyone could give me any help on this that would be much appreciated. I have no idea where to go or what to do.