I assume you are looking for an intuitive explanation rather than a mathematical proof. I find it easier to think about fictitious play FP than CFR.
- Player 1 finds best response (BR) against player 2. Player 1 strategy becomes average of BR and old strategy
- Player 2 finds BR against player 1. Player 2 strategy becomes average of BR and old strategy
- Repeat

The nash equilibrium cannot be a best response because BR is pure and exploitable. The averaging process ensures that given enough iterations the right mix of pure BR is built up to provide the best possible defence. On each successive iteration the change to the average strategy gets smaller. Work through rock, paper scissors on paper to illustrate it to yourself. Hopefully you could apply the intuition found in FP to CFR.

There are well known abstractions, but you can still use your own. I usually use a modified EM abstraction.
You will probably need to use an abstraction, except if you only want to solve the river or turn. If you want to solve the whole game, you will need card abstraction.

The reason why CFR+ isn't well suited for abstractions is, that most of the state of the art ones are imperfect recall. So cards that are in different buckets on the flop, might be in the same bucket on the turn again. I think it can probably be done with CFR+, but you would somehow have to map the probabilities for each bucket back and forth on every new street, which is slow and hard to code. And even that I am not even 100% sure it would work.

It's not necesasrily the best algorithm, for instance it's not easy to use if you want to use abstraction techniques, in that case Monte Carlo methods are probably better.

If you don't want to use card abstraction, CFR+ is probably the best algorithm. It can use averaging though, the average still converges faster than the regrets. The reason why averaging is often (not always) ommited, is because you basically only need half the RAM if you don't have to keep track of the average strategies. That is important in cases like solving the full FL game, but if you have smaller games and you have enough RAM anyway, I would still use the averages.

I assume you are looking for an intuitive explanation rather than a mathematical proof. I find it easier to think about fictitious play FP than CFR.
- Player 1 finds best response (BR) against player 2. Player 1 strategy becomes average of BR and old strategy
- Player 2 finds BR against player 1. Player 2 strategy becomes average of BR and old strategy
- Repeat

The nash equilibrium cannot be a best response because BR is pure and exploitable. The averaging process ensures that given enough iterations the right mix of pure BR is built up to provide the best possible defence. On each successive iteration the change to the average strategy gets smaller. Work through rock, paper scissors on paper to illustrate it to yourself. Hopefully you could apply the intuition found in FP to CFR.

If we use CFR to calculate our regrets for each decision and then recompute strategy based on those regrets. After billions of iterations, why is the final strategy that we used not one that converges to the nash equilibrium? It's said instead that the average of the strategies do

The reason why the average strategy converges and regrets don't is basically just math. There is a proof in either the Polaris or Hyperborean paper by the University of Alberta.
CFR+ regrets often converge to a Nash Equilibrium, because they use a different form of regret matching. I think noone really knows why it's so much better, they just found out by testing it.