Things start to seem right to most of us when we get to the discoveries of Newton. We may not know what attracts objects to each other but we don't mind just knowing that it's called gravity. We like the idea of an orderly predictable clockwork universe in which everything moves along in cosmic harmony.
Maybe the Newtonian view of the world feels comfortable because it's still within the realm of our senses. Our common sense is based on the world around us. We feel comfortable dealing with ideas that are within our grasp. You pick up an apple and let go and it drops towards the centre of the earth. There's nothing unusual happening there. You throw the apple hard and it moves off at a certain speed. Throw it twice as hard and it leaves your hand moving twice as fast. That's practically intuitive.
Then along comes Einstein and the concepts don't mesh well with our intuitive sense of the world around us. It's hard to imagine that someone on a moving train or rocket ship has a different perception of time than you do. You see them moving slower and they see you moving faster. All because there's an absolute speed limit and light has to always move at that speed. Light can't travel through a vacuum slower or faster. No one can see light travelling through a vacuum slower or faster than anyone else.
We may grasp the idea that space is curved but we grasp the concept using the metaphor of a rubber trampoline that dips whenever heavy objects are placed on it. It's much harder to visualize a three dimensional universe as having curves near large objects.
It's even harder to grasp the idea that since objects have a potential speed limit you can't just keep throwing them harder to make them move faster and faster. As you push something faster and faster it gets harder to speed it up. Let's say you take one of Newton's apples and push it from a standing start to one quarter the speed of light while measuring how hard you had to push. Now take the apple from a standing start to half the speed of light. Turns out it takes more than twice as much pushing.
So where does that extra energy go? If it takes more than twice the push to go twice as fast what happens to the extra push? That's where Einstein's famous equation comes in. Energy and Mass are equivalent. The extra push goes into making the apple more massive. Yes... the reason it gets harder and harder to push apples or rocket ships is that they get heavier and heavier the faster they go.
This isn't within the realm of our common sense. It makes you want to go back to Newton's universe and sit under an apple tree doing simple calculations.
What happens when you move on to quantum mechanics?
As I see it there are two choices. You either give up and say it's not understandable by the typical person or you go looking to see if someone has attempted an explanation that makes some sense.
One such attempt was made in 2008 by Eliezer Yudkowsky on Less Wrong. A series of posts called the Quantum Physics Sequence goes through many of the concepts behind quantum mechanics and does a respectable job trying to make them more intuitive.
The entire sequence of articles is in the link above. There are also a few short lists that take some of the articles and help you understand part of quantum mechanics. Quantum Physics Revealed As Non-Mysterious is a good look into why modern physics isn't all that hard to grasp.
Eliezer's central approach can be summed up by a quote from the article in the sequence called Quantum Explanations:
...I don't believe in phenomena that are inherently confusing. Confusion exists in our models of the world, not in the world itself. If a subject is widely known as confusing, not just difficult... you shouldn't leave it at that. It doesn't satisfice; it is not an okay place to be. Maybe you can fix the problem, maybe you can't; but you shouldn't be happy to leave students confused.
The first way in which my introduction is going to depart from the traditional, standard introduction to QM, is that I am not going to tell you that quantum mechanics is supposed to be confusing.If you don't want to give up on trying to grasp quantum mechanics then give the Quantum Physics Sequence a read. You'll be surprised how much of it you can grasp.