Here's an attempt at a more intuitive explanation that makes things feel pretty clear to me.
Let the circles be $A,B,C,D,E,$ read from top to bottom and left to right. We're going to move $A$ righwards, $B$ upwards, $C$ downwards, and $D$ and $E$ leftwards, all at a constant rate (though not the same for every circle). We'll argue that this never causes the circles to intersect, so we can continue it until one of them hits a side of the rectangle.
First, let's assert that $D$ and $E$ move left at unit speed. We'll try to work out how fast everything else should move.
Consider the half-plane bounded by the tangent between $B$ and $D$ that contains $B$. If we move this half-plane left at the same rate that $D$ is traveling, this is equivalent to moving it up at some other rate related to the slope of the line. So if $B$ moves upwards at that rate, it'll stay within a half-plane that doesn't overlap $D$ at any point.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/AH1P6.gif)
Likewise, we can take the half-plane bounded by the $C-E$ tangent which contains $C$, and note that moving it left is equivalent to moving it down, at the same rate as in the previous paragraph by symmetry. So we can also move $C$ downwards at the same speed $B$ is moving upwards.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/N3s41.gif)
Now consider the half-plane containing $B$ and bounded by the $A-B$ tangent, and the one containing $C$ and bounded by the $A-C$ tangent. These will move up and down, respectively, at the same rate, which means that they can be thought of as moving left and right, respectively, at some other rate (but still the same one between the two of them, because they meet $A$ at identical slopes). So, by the same logic, the void between these half-planes will keep having space for $A$ as it shifts rightwards at some third constant speed.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/nPG67.gif)
Here's a gif of all five circles moving at a constant speed from one end of the square to the other, with a single line for each relevant half-plane. Note that every line touches the corresponding two circles at all times, and all that changes is the relative position of those tangent points.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/RPPzF.gif)