There seem to be a lot of misconceptions and outright ignorance about Common Core here. Common Core is basically just a restructuring of when different subjects are introduced, and how much emphasis is placed on each area at each grade level. For example, in mathematics where previously you might have an algebra class one year, then a geometry class another year, then trigonometry another year, etc., this might get reorganized so that material from each of these courses is introduced at different times in what proponents claim is a more logical structure that achieves better results (and there does seem to be a lot of evidence to support it). So instead of Algebra in 7th grade and Geometry in 8th, you might get some parts of what was in the Algebra class in 6th grade, a little more in 7th, some more in 8th, while also being introduced to Geometry earlier and having that spread across multiple years. You end up in the same place (well, hopefully on average you end up a little more advanced by the end), but at any given point in their schooling students will be ahead of where they would have been under the past system in some areas, and behind in others - by design.
However, this rearrangement of coursework opens a can of worms, which is where most of the fighting comes in. Because things are introduced at different stages and in a different order, an entirely new curriculum is required. It is left to the states to decide what curriculum to use, and there are a lot of choices - much of it produced by commercial entities, some of it good and some of it really, really bad. This isn't a function of Common Core, per se, but merely a function of lots of groups taking advantage of a major re-write to try to get their product included in what is selected at the state or local level.
Likewise, since the order things are introduced changes, all of the standardized tests are no longer relevant - children might be learning some of what falls into "algebra" in the current system in the 5th grade, so a standardized assessment test would need to take this into account. Opponents latch onto this and complain that too much is expected of the students, because they are being tested on something "too advanced". Likewise, something that students previously learned in the 4th grade might not be introduced until the 6th - and again, opponents latch onto this because the standards have been "lowered". It's easy to cherry pick examples that go either way (which this comment section is rife with), because compared to what most of us experienced, it will feel "off".
The vast majority of the arguments against Common Core aren't actually about Common Core, rather they are about some of the curricula that have been developed to meet Common Core's structure. Just like there can be a fight every time a new science textbook is chosen in Kansas (or anywhere else), everyone is arguing over what the curriculum should look like, and it is all happening at once. So, lots of people trying to get their own political slant into the new curriculum, which is the same problem as always - it's just happening all at once across pretty much every subject.
Now, there are certainly objections or questions to ask regarding Common Core. For one, are the benefits of the rejiggering of subjects enough to outweigh the costs of introducing the system? What do you do about students who started with one system - can you transition them to the new standards effectively, or will we have several years worth of students with glaring holes in their education? And last (and probably the biggest question, and the one that has driven many one-time supporters to oppose common core), how do we ensure that the curriculum chosen by my school district/state/whatever is going to be effective and not just an amalgamation of commercial offerings selected through a combination of ideology, lobbying, and kickbacks - the educational outcomes are dependent on the effectiveness of the curriculum, and there is no guarantee that new ones being developed and offered will achieve that (and, for the reasons mentioned, a lot of reasons they might not).