Intuitively, $\theta$ represents longitude on the (hemi-)sphere, and $\phi$ represents co-latitude. If you let $a$ vary, then it would describe altitude.
To see that these coordinates actually describe the hemisphere is straightfoward: all you have to do is check that $\begin{align*} x & = a \sin \phi \cos \theta \\ y &= a \sin \phi \sin \theta \\ z &= a \cos \phi \end{align*}$ actually satisfy $x^2 + y^2 + z^2 = a^2$.
But how would one actually derive these coordinates? Essentially, it works the same as in 2-dimensional polar coordinates: draw the right triangle and use "soh cah toa."
So which right triangle do we draw? Let $O$ denote the origin, let $P = (x,y,z)$ denote the point on the (hemi-)sphere in question, and let $Q = (x,y,0)$ denote the projection of $P$ directly down onto the $xy$-plane. Look at the triangle $\Delta OPQ$. Mathworld's page on spherical coordinates has an illustration of this.
The height of this triangle (side $PQ$) has length $z$, the base (side $OQ$) has length $r = \sqrt{x^2 + y^2}$, and the hypotenuse (side $OP$) is the radius of the sphere $a$. Then by "soh cah toa," $\begin{align*} r &= a \sin \phi \\ z &= a \cos\phi. \end{align*}$ Now we examine the point $Q$, and use 2-dimensional polar coordinates. Since the length of $OQ$ is $r = a\sin\phi$, we have $\begin{align*} x &= (a \sin \phi)\cos \theta \\ y &= (a \sin\phi)\sin\theta, \end{align*}$ which is what we wanted.