There are many ways to explain this, but it's hard to spell out the rules (for English modal verbs) precisely. I'd like to give you a little tip to deal with them:
In English, there are two main time domains (non-past and past), and thus there are two verb forms for them: the present form and the past form. (Let's not discuss the past participles for now.)
The main uses of the past form are a) for things that happened in the past, and b) for things that didn't or don't actually happen (such as our "hypothetical race" in this video).
The following is disclosed-captions™ (mainly for the times and tenses) for the above video clip:
How could a humble tortoise beat the legendary Greek hero Achilles in a race?
This is a typical use of could. It's commonly used as an opening question of presentation, a talk, a speech, and such. This one invites the listener to take it as a thought experiment.
The Greek philosopher Zeno liked the challenge and came up with this paradox.
There is nothing special here. Zeno lived 490 BC - 430 BC. At the time you listen to the video, all Zeno's actions happened in the past, so we use a past tense: the simple past.
First, the tortoise is given a slight head start.
Now the narrator is telling us about a "rule" of the race (which is our thought experiment). If he used the simple past instead of the present simple, it would be from Zeno's perspective, but he simply uses the present simple which makes it easier for us to get along with what he's telling. It's more vivid this way, as if it were really happening in front of us. (And the video reinforces this effect!)
Anyone fancying a flutter would still rush to put their money on Achilles.
This would is used because it's hypothetical. If the race were about to happen now, those who wanted to make a bet would bet on Achilles. They would think Achilles would win the race.
But Zeno pointed out that, to overtake him, Achilles would first have to cover the distance to the point where the tortoise began.
The narrator tells us that Zeno pointed that out (in the past), that if there was such a (hypothetical) race, Achilles would (hypothetical) have to do that, he would have to run to the point where the tortoise began (hypothetical). This hypothetical race is timeless. In other words, whenever we had such a race, if we could arrange such as a race, Achilles would have to do that, every time.
In that time the tortoise would have moved – so Achilles would have to cover that distance, giving the tortoise time to amble forwards a bit more.
We are still in the hypothetical race here, but notice the use of would have moved. It's not just would move. It's would have moved. The use of would have moved tells us that if that hypothetical race is a real one, the tortoise will (logically) have already moved. It has already moved. And because the tortoise has already moved, Achilles can do nothing better than to cover that distance. So in our hypothetical race, he would have to cover that distance.
Logically this would carry on forever.
Logically, our hypothetical race would keep going on and on, forever!
However small the gap between them, the tortoise would still be able to move forwards while Achilles was catching up, meaning that Achilles could never overtake.
The use of the past forms in would still be able to, was catching up, and could never overtake is only because the race is hypothetical.