Bridle cheeks buckle through the small slot immediately adjacent to the curb hooks. Your pic shows it upside down.
Reins can be fitted through either slot or simply buckled around the ring so they slide up and down which was the original idea - you dropped your hands to get more curb action or raised them to get less.
This bit is more correctly called a 'Kimblewick' in the UK - it's named after the village where it originated. The model with slotted cheeks is an 'Uxeter Kimblewick' (or Kimberwick
if you must).
They used to be popular for ponies, which could be strong for a young rider, because it provided a little of the action of a pelham, but without the difficulty of managing two reins or the potential for harm in inexperienced hands.
They fell out of favour in the 1980s with the introduction of the frankly awful 'Dutch Gag', which can cause far more damage IMO.
We last used a Kimblewick on a pony who had the habit of charging out of the schooling ring in a snaffle. After wearing the bit for a fortnight she went back into a snaffle and had no more issues with running off. Cavalrytales Blog