How can some theories claim the existence of higher dimensions? Note: The following question is probably demonstrating a complete misunderstanding of higher dimensions. I am by no means an expert in string theory and higher dimensions. I'm a 7th grader who recently wanted to learn about the 4th dimension. I would ask that you please try a bit to keep this in mind in your answer.
Recently, I have learned that superstring theory claims that there are 10 spatial dimension existing in our world. How would that be possible in a 3 dimensional universe? If there is an actual "4th dimension" out there, it would have to exist as a subset of a 3 dimensional universe, which simply does not make sense. Or, you could be saying that it exists outside our universe, which means there is an edge to the universe, or at least the 3D part of it, meaning you could stick your hand out and be in two dimensions at once? That also doesn't make any sense. So, how can string theory claim there are more than 3 spatial dimensions in our world?
 A: 
I have learned that superstring theory claims that there are 10 spatial dimension existing in our world. How would that be possible in a 3 dimensional universe?

If the universe has 10 dimensions, then it is 10-dimensional and only appears 3-dimensional. The other 7 spatial dimensions would have to be very "small" dimensions. An analogy would be a sheet of paper, which is 3-dimensional but appears 2 dimensional because one dimension, it's thickness, is very small compared to the others.

If there is an actual "4th dimension" out there, it would have to exist as a subset of a 3 dimensional universe, which simply does not make sense.

Actually, the 3-dimensional universe would be a subset of the 10-dimensional universe and "fit inside" it, the way a line fits on a piece of paper and a piece of paper fits inside a box.

Or, you could be saying that it exists outside our universe, which means there is an edge to the universe, or at least the 3D part of it, meaning you could stick your hand out and be in two dimensions at once?

There doesn't have to be an edge to anything, so long as the dimensions in question are unbounded. You can imagine an infinitely long line, for example, which has "edges" (boundaries) in two dimensions but not in the third. That line could have two similar lines orthogonal to it, and orthogonal to each other, and each of those infinite lines could act as the axis of one of our three dimensions in space, which also would then be infinite. Closed shapes can also be finite without having an 'end' or 'edge:' where is the edge of a circle, for example? If you think about it, when you stretch out your hand, you're not just in two dimensions at once, but three: but even though each of those three dimensions has "edges" of its own, in that it is only one dimension itself, the universe doesn't have an edge... That we can see at least.
On the topic of "curled up" dimensions, I'll give another analogy: if you've ever played the classic arcade game Asteroids, (check it out at freeasteroids.org if you haven't,) you're familiar with the concept of a small, closed dimension: when you go a certain distance in one direction, you come to what looks like an edge, but its not really an edge because you wrap back around to the other side of the screen. The left "edge" is connected to the right "edge," and the top and bottom "edges" are also connected to each other. Each of the two dimensions in that game is a small (only a few inches) curled dimension. Your ship can travel in the same direction forever without ever leaving your computer screen. Now, imagine that in the real world there are closed dimensions like that, but instead of being a few inches like the Asteroids game, each dimension is about the size of subatomic particle like an electron: that's what string theory supposes the other seven spatial dimensions we can't see are like.
A: Why was science before relativity  postulating three space dimensions and one time?
Because that was the distillation of observations over millennia.  Once geometry was established as a mathematical discipline/model it was the simplest way to describe/fit observations. 
What are observations/measurements?
They are the way our five known senses interact with the environment and bring information to our brain, which organizes it for retrieval. Writing and arithmetic extended the organizational pattern so that information could build up over generations.
The technological advances after the 18th century increased the possibility of observations and the advances of mathematics increased the complexity of fitting the observations/data with the aim of having as simple a mathematical model as possible.
Dimensions kept being 3 space ones and one time, the space ones a vector space with a length well defined for vectors.
The data began to be complex enough that in the beginning of the 20th century special relativity was needed to describe the data, and it needed time to be part of a pseudovector space with a length defined in this space. Special relativity fits the observational data extremely well, in nuclear physics and particle physics.
In parallel the forces observed acting on particles were organized in a coherent manner in a coherent mathematical model . These forces at the particle level are interactions of point particles with each other, and complicated mathematics describes accurately the observations. Then the search of physicists for unifying theories, i.e. mathematical models that unified all the interactions/forces, led to higher dimensions.
As you state we do not see with our five senses any higher dimensions. Theorists are smart though, so in the theories they postulate ( like axioms in mathematics) that the other dimensions cannot interact with the particles we have observed and classified, except through the gravitational interaction.  Their smartness extends to postulating that those extra space dimensions are curled up, like little circles going around, in so small dimensions that we would never see them even interacting gravitationally with them. This, because we have not seen particles disappearing and leaving four dimensional conservation laws hanging/violated.

meaning you could stick your hand out and be in two dimensions at once?

Curling up the extra dimensions means that you and I and everything , even though we exist in those, our hand will never interact and tells us that they exist, as it tells us that a third dimension exists outside a window.

That also doesn't make any sense. So, how can string theory claim there are more than 3 spatial dimensions in our world?

The claim is a well formulated theoretical hypothesis which is being tested with higher and higher energy experiments at LHC at CERN  . There are predictions of superstring models for the experiments at LHC, and if the predictions pan out then one can take the higher dimensional theories seriously and push for more and more experiments to see what is up.
