They don't choke them, they sever their spine, and will hunt much like a large tiger.
The first thing we have to take care of is the beak: most felines today don't exclusively use their claws to end prey, and thus the head will need to be of use in the taking down part. You're right that a beak is normally not what you think of when you think of a pair of mandibles used to choke a creature. However, when it comes to birds of prey, one of the best examples of a raptor that relies on its beak is the peregrine falcon.
The triangular ridge you see near the tip of the falcon's beak is called a tomial tooth. These structures are needed for falcons because, unlike other raptors which rely a lot more on a crushing beak and long talons to kill prey, falcons use their beaks, adapted specifically to bite the prey's neck and sever their vertebrae, killing them fairly quickly. Of course though: this thing would need a slightly modified head with powerful jaw muscles to make sure the sharp beak can actually do the slicing, which makes a large head a good thing (the larger it is, the more space for muscle it has).
However, there's always the problem that the falcon isn't normally dealing with: having its prey consisting mostly of other birds and small rodents, it's beak rarely has to deal with biting a neck with a higher level of protection or that belongs to a larger struggling creature. On such a scenario, we have to step up the game, at which point your demigryphon might need a beak and jaw structure more like those of an alligator snapping turtle. 
The alligator snapping turtle is well known for 2 things: its fast and powerful bite and it's sharp, pointed beak, and while the falcon uses its beak more than other raptors, the turtle uses its jaws exclusively, being completely dependent on that very beak and powerful muscles to bite down on the fish it lures. The biggest differences in this arrangement are the more straightened, pickaxe-like tips of the alligator snapping turtle and its more robust jaw muscles, which when scaled up accordingly, could potentially allow your demigryphon to perform a more active stabbing movement like what we see in big game hunters, with the "hooked" tips of its beak playing the role of canines digging into the flesh (also, if thylacoleo, the marsupial lion, is something to go by, it's still possible to perform the stabbing part with more frontal, beak-like structures while having a hunting style closer to that of a modern large feline). On such a scenario with a more robust and sharper beak, as well as more powerful jaw muscles, we could, in theory, have a scenario where your demigryphon could still employ a vertebrae-crushing strategy on smaller prey, but the longer pointed extensions at the tips of the beak and powerful bite could probably make room for a feasible, although maybe not as efficient, throat grab strategy, with the powerful muscles handling the constriction of the airways while the hooks deal with puncturing vital blood vessels and ensuring a proper grip onto the throat.
Now, with the beak part dealt with, how do they hunt? Given these things have literal bird heads, I'd say they probably have similar structure and brain development in areas closer to those of a predatory bird than to those of a feline. This means sight and hearing will probably be the most important things for it, as many raptor species have these senses more well developed than their sense of smell and taste.
Basing myself on terror birds, the most notable example of large terrestrial predatory bird we have, the demigryphon would probably depend on keen sight and a great sense of hearing, particularly adapted for lower frequencies to track prey, meaning that it'd rely more on hearing its prey moving around and spotting it from far away than actually tracking it through smell. The raptor eyesight would make this thing pretty good at spotting creatures from an advantageous position, and the low frequency hearing would mean that it could both hear prey well and communicate from large distances, since lower frequency sounds travel further than higher frequency ones.
Regarding what it'd do when spotting prey, it's pretty obvious: its body and size implies it would hunt much like modern Bengal tigers in terms of strategy and behavior (funny enough, these big cats also rely more on sight and hearing than smell), as in it probably would hunt as a large ambush predator only capable of short distance pursuits at best, using its powerful limbs and large size to tackle prey to the ground and target the back of their necks with its powerful and sharp beak. Its large size would also allow it to scare off other smaller predators and scavengers, granting better access to carrion and free food when given the chance. Furthermore, its large size compared to the size of available prey probably means it'll also have to be solitary, staying away from others of its kind outside of mating season. An ill demigryphon will probably behave much like a man-eating tiger, it'll hunt humans out of desperation due to being ill or injured, and thus unable to hunt its normal prey and needing to go for smaller, slower game.
As for what it'd hunt, again we look at tigers. Its preferred prey would probably be large ungulates which live in areas of dense vegetation where it can approach them more easily, while staying away from larger animals it can't face on its own. More desperate demigryphons in regions with reduced big game numbers would go for smaller prey such as deer and wild boars, and very desperate/ill demigryphons would attack villages and eat people in there. It's also not crazy to consider them targeting larger livestock, since those are large, slow moving, AND cornered (if it does have the same capabilities of a Bengal tiger you know it's deceptively agile, even though it's not the speediest around). Demigryphons in regions with less vegetation might also be inclined to form hunting groups for a higher chance of success when hunting at the cost of smaller, although more frequent meals.
