What is the point of this stance? Eagle kung fu 'Eagle claw' stance?

Besides the training and conditioning of the hands(fingers) to be like the eagle talons, the movements too should assume some 'bird-like' postures. That is the romantic aspect in CMA. The three types of 'claws' in CMA: tiger, dragon and eagle, the latter is most difficult to master. As to the stance, with one leg on the thigh of the other, it is not a dynamic stance but static. If you have watched Jet Li + Jackie Chan's Forbidden Kingdom, you would have seen Jet Li in the same poise. In the Northern Shaolin repertoire, there is a similar poise but with both hands clasped in a prayer poise. So perhaps, for those in this forum who is looking for artistic "nonsense" and give credence to trolls, this is it.

l'm no expert in Eagle Claw Kung Fu, but it looks like a takedown technique to me, kinda like a variation of the reaping takedown with your right hand spread like that probably because it's redirecting the opponent's sword arm and and your left leg hooking his leg throwing him off balance. You must remember, the style as the article states, was allegedly created by a Chinese General who fought during the Song Dynasty period. Most warriors during that time carried weapons and most fighting styles were weapon based. If you notice the bottom picture where the practitioner performs the split, this technique is also found in sword style forms where the practitioner avoids an overhead strike by performing a split and striking the opponent's belly or groin.

Without seeing the beginning movement, I would have to say that it adds gravity power to a technique, with a bottom vector. One leg and 2 arms would have to be doing something useful however, but without a body to hold unto, it would just be floating in mid air.

The core foundation of the static stance is that the weight is dropped directly down (there's only one leg there, it can only go down without spinning). There's a number of techniques that can be made more efficient with extra downward force. It doesn't need to be a specific one.

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that stance is not really in my style. although it part it reminds me of a namiashi technique in naihanchi.

stances are dynamic techniques they are not fixed, they are part of your techniques, besides just bring balance they are part of grappling. what its for depends upon the preceding technique as well. it could be for something as simple as avoiding a leg sweep while striking, as well as a strike it self. or it could be far more complex such as a lock and trap inside the stance

To make yourself as vulnerable and as unstable as possible so you can get your butt kicked by an average fighter.

It makes you more flashier in a fight, making the crowd cheer you even more and decreasing the morale of you enemy. :)