What is this old "lemon-squeezer" shaped pan
What you have appears to be a vintage or possibly antique cast iron grill pan.

The pan would be used on the stovetop and could be used for steaks, chops, burgers, bacon, certain vegetables, etc. The ridges can provide grill marks, and the channels allow grease to drain away. The shape allows the grease to drain to the outside channel where the pour spout will allow the grease to be poured out of the pan.

As with the grill pans of today, the pan would be pre-heated and the meats or veggies would be cooked at a medium high or high heat, much like outdoor grilling in the US. Also, it would be good for camp-type cooking over an open flame.

Re rendering fat, I suppose it could work but with a low or slow heat. 

Update: As per @Sneftel's comment, these grills can have many different uses. @Falken's answer addresses one of these, but certainly not all. In addition to the uses mentioned previously, see this snippet from WebstaurantStore:


  An all-around convenient stovetop grilling solution, this 10 1/4" round heavy-duty cast iron barbecue plate is perfect for making a variety of delicious items including Korean barbecue, pita bread, naan, grilled tortillas, and roasted vegetables.


Similar pans can easily be found online by searching for Mongolian bbq grills.