If $A$ and $B$, are subsets of $\mathbb R^n$ then you can map $A\times I\times B$ to $\mathbb R^n$ by $(a,t,b)\mapsto (1-t)a+tb$. If the resulting continuous map $A*B\to \mathbb R^n$ happens to be one to one then you can ask whether it gives a homeomorphism to its image. If $A$ and $B$ are compact, then the answer is necessarily yes, since a continuous bijection from a compact space to a Hausdorff space is always a homeomorphism.
This suffices to see that the join of spheres is a sphere.