What you must remember in regards to Klingons is that the ones we're used to seeing are from a time further in the future, or even the past, and the ones dealt with in the past had very shaky practices. We are also sitting in a setting of the Star Trek universe where Klingons have attempted to go back to their old ways, and like with real world religions and historical icons, they may have misinterpreted it. You can't compare current timeline Discovery Klingons with the Klingons from the TNG/DS9/VOY universe ( though even the Voyager Klingons were different than the ones from TNG/DS9 due to having been in the Delta Quadrant as a generational ship for so long ) because it is a 200 year difference in their culture.
What we're likely going to see develop over the course of the first season is the backstory as to how they've gone back to such a highly aggressive state, something that looks like it's being pressured by a fanaatic with a lot of power, and the eventual turnaround to set Klingons on the path we know from the TNG timeline.
It may also explain the dramatic visual change as well - at some point during ENT with the Augment virus being used and altering the appearance of over 1 million Klingons and their descendents to follow, the Klingons unaffected may actually end up wiped out or equally dramatically altered during this. There's also the potential of them having altered their own appearance through cranial reconstruction and, as fanatics are wont to do, going too far.