This is just a normal pullback square in some category, arranged a bit differently:
a & \leftarrow & a \times_c b & \rightarrow & b \\
& \searrow & \downarrow & \swarrow \\
& & c
Now in the slice over $c$, each arrow to $c$ gets turned into an object. Let's call them $f$, $g$ and $h$ from left to right in the above diagram. The arrows $a \times_c b \rightarrow a$ und $a \times_c b \rightarrow b$ each make a triangle with two of the arrows $f$, $g$ and $h$ commute, so in the slice we have arrows
$$(a,f) \leftarrow (a \times_c b, g) \rightarrow (b,h)$$
which are projections making $(a \times_c b,g)$ into a candidate for the product $(a,f) \times (b,h)$ in the slice. Now the universality of $(a \times_c b,g)$ can be seen by adding in another candidate for the product in the slice and transfering that into the diagram in the original category, where it becomes a candidate for the pullback $a \times_c b$.