Basically you need the "infinite/zero" point to compensate for a situation when you have two points completely perpendicular to the x-axis. AKA it is not intersecting a third point. So it intersects this special "infinite" point.

And conceptually why you need this "infinite" point is that without it you can't add points together properly.

Say for counter argument instead of doing this "flip or mirror" across the x-axis (in the interaction it is the red dot appearing). And instead the red dot just appears on the same side as the two points being added on the curve - without the flipping.

If P1+P2 = Q instead of this Q' that is flipped. And P2+Q = P1

If you try and add P1+P2+Q you would get either Q+Q or P1+P1 depending on if you did (P1+P2)+Q or adding up P1+(P2+Q) which are not equal.

so you need this red dot flipping thing happening in the interaction. However, if you have this flipping that means P1+P2 = Q' which is the mirror flip of Q.

So Q'+Q need to equal this special infinite/zero point to ensure associativity works.

But AFAICT, elliptic curve groups really are the best known groups where DH is hard. The "Why curves win" section talks about it terms of key size, but the reason other groups require larger keys is they have some kind of structure which can be exploited to attack the "hard" direction (eg. in a finite field, the ability to factor over primes can be used to solve discrete logs), so the group size has to go up to compensate.