One way of thinking of it is in terms of how far along the rope the ant is. If the ant is 20% along the rope, and the rope stretches, the ant is still 20% of the way along a now-bigger rope. Therefore the ant never loses ground, so to speak.
The effect of the stretching is to reduce the ant's speed measured in units of "proportion of the rope". E.g. if the rope is 100cm and the ant is going 1 cm/s, then the ant is covering 1% of the rope per second. If the rope doubles in length and the ant keeps the same speed, the ant is covering 0.5% of the rope per second. So the ant is decelerating in this set of units; the total distance it travels becomes a matter of computing a limit.
> But this will always be a positive number, therefore in some finite number of seconds, the ant will cover 100% of the distance.
Correct result, but that reasoning is dodgy.
It's easy to imagine a different problem where an ant always covers a positive proportion of the rope every unit time, but the ant doesn't reach the end of the rope. E.g. construct a situation where, in the 1st unit of time, the ant covers a quarter of the rope, in the 2nd an eighth, in the 3rd a sixteenth, ..., in the nth a 1/2^{n+1}th. In that scenario, as time tends to infinity the ant asymptotically reaches the half-way point.
The reason it works in the problem in the article is that, there, the additional proportions the ant covers in each unit of time form a divergent series (~the harmonic series), meaning the sum tends to infinity as n-> infinity. In my version above, they form a convergent series, meaning the sum tends to a finite number (in my example, 0.5) as n-> infinity. (Of course, a convergent series with a sum >1 would also mean the ant reaches the end of the rope in finite time).
The effect of the stretching is to reduce the ant's speed measured in units of "proportion of the rope". E.g. if the rope is 100cm and the ant is going 1 cm/s, then the ant is covering 1% of the rope per second. If the rope doubles in length and the ant keeps the same speed, the ant is covering 0.5% of the rope per second. So the ant is decelerating in this set of units; the total distance it travels becomes a matter of computing a limit.