Today in physics class, we learned about the theory of relativity and how it relates to time dilation. We were given the example of two photon clocks- one stationary relative to the Earth, and the other on a rocket going at a very high velocity relative to the Earth. The moving photon has more distance to travel than the stationary one, and because the speed of light is constant for all observers (I don't understand why, but from what I understand, no one knows why for sure), time must slow down for the rocket clock in order for this situation to work. This makes sense to me, and I now understand the notion that higher velocities can slow down time. However, if we flip the scenario, and make our stationary frame of reference the rocket, the photon clock on the Earth is moving at a negative velocity relative to the rocket, and the photon on the Earth must travel a greater distance, which means that time must slow down the same amount for the photon clock on the Earth relative to the rocket as the clock on the rocket relative to the Earth. This breaks my understanding, as it seems to me that time is slowing down the same amount for both, but I know that only the clock on the rocket is slowing down from an absolute perspective. How is this possible?
It doesn't, and you are half-mistaken.
That is to say: yes, if Alice is moving at speed $c/2$ relative to Bob, she thinks Bob’s clock is ticking slowly. Yes, Bob also thinks that Alice’s clock is ticking slowly. (That is not the same as what they see because one also had to factor in the Doppler effect; so if they are moving towards each other then they both see each other's clock ticking fast, albeit not as fast as it “should be ticking.”)
Where you are mistaken is thinking that one of them is objectively right. Neither one is.
If you tease out why you think one of them has to be right, you will probably come to a question about their mutual interaction. So let's say you are Alice, you might be frustrated enough to say “I hate all of this! I am just going to call Bob up and one of us will be talking slower than the other!”
Ok. So you use a cell phone. Those transmit microwaves which travel at the speed of light. So you say something to Bob at first and it takes a while to get to him. Then because he is coming towards you there is this Doppler effect: he is speaking very fast, but not as fast as you would have expected. But he sees the same thing: the time gap resolved the paradox.
You want an infinitely fast communication medium? Ok, then that will force Nature to either pick a side, or to allow information to travel backwards in time. But, curiously, we haven't found one yet. Nature solves these paradoxes by not giving us the tools to realize them.
It is also worth mentioning that this time dilation effect is second-order in velocity, which means that you can derive it from the rest of relativity. The only first-order effect is that when you accelerate in some direction, clocks ahead of you by a distance $\ell$ tick faster by a factor $a\ell/c^2$ and clocks behind you tick slower by the same factor. This causes an offset that we call “the relativity of simultaneity.” But length contraction and time dilation are just an accumulation of these moving disagreements of what time is “now” at far-away points. So for example in the twin paradox it becomes very important that one twin accelerates towards the other twin, as during that time they see their twin's clock ticking extremely fast, solving the paradox of which twin is really older. But as for why the universe is arranged in such a way that accelerations work this way, that is hard to understand and may ultimately come from some very-strange explanation like “maybe everything moves at the speed of light intrinsically, and we only see things appear to move slower because things bounce around off of each other a whole lot.”