Why are antibonding orbitals higher in energy?
You shouldn't stop asking "why," but really explaining why is really, really complicated. It sort is like learning to run. First you learn to waddle around, then you can walk, and then you can run. Starting off such an explanation with quantum mechanics isn't throwing you in the pool it is more like throwing you in the middle of the Pacific Ocean.

Start with this. I assume that you're happy with the shell structure of atoms. Now we want to make molecules. 

Consider the reaction: $$\ce{H + H -&gt; H2}$$

Two hydrogen atoms each with a $\ce{1s^1}$ electron bond to form a "molecule" with 2 electrons in some sort of orbital. There is no shell model for a molecule, so how do we conceptualize the two hydrogen orbitals combining? A straightforward approach is the linear combination of atomic orbitals (LCAO). So we have a $\ce{1s}$ orbital from the first hydrogen and a $\ce{1s}$ orbital for the second hydrogen. Remember that each of these orbitals can hold two electrons. We now, with some nativity, just assume that the energy of atomic orbitals must be conserved. So how much in energy the molecular orbital is lower than the hydrogen orbital then the other orbital must exist with that much "antibonding" energy. But for $\ce{H2}$ there are only two electrons, thus the antibonding orbital is empty and the net result is that a $\ce{H2}$ molecule has less energy than two lone H atoms and the molecule will stay together. 

Now consider the reaction:
$$\ce{He + He -&gt; He2}$$
Both He atoms have two $\ce{1s}$ electrons. Thus when we try to form a $\ce{He2}$ molecule both the bonding and the antibonding orbitals are occupied. The net result is that the atoms in the $\ce{He2}$ molecule have no net affinity to stay bonded together. Notice also that this would not be true for a $\ce{He2^+}$ ion. With two electrons in the bonding orbital and only one in the antibonding orbital the ion would be "stable" in free space - at least until the ion bumped into some other atom or molecule in interstellar space. 

Now using LCAO is naive. However it works well for a lot of chemical reactions. The really nice thing is frankly that the mathematics is fairly simple. Think of Newton's gravity equation. It worked well enough to put men on the moon, but it isn't exact. Now with GPS satellites the GPS clocks must be corrected for relativity. Scientists knew about relativity before we went to the moon of course, but the expanded mathematical treatment just wasn't necessary. 

So does that answer your question in enough detail so that you can accept the explanation?

Edit - adding information about quantum mechanics

The easiest treatment of atomic orbitals is to assume hydrogen like orbitals and tweak the value of the orbitals based on the interactions of electrons. The gist here is that the simplest mathematical treatment assumes that the electrons move around the nucleus in defined orbits like the planets move around the sun. It was quickly realized that this sort of modeling was woefully inadequate. 

Even with this simple treatment there is a massive fundamental problem that bedevils all such calculations. The three body problem in physics. If our solar system just had the sun and the earth, then it would be possible to use Newton's gravity formula to solve the equation which predicts the movement of earth exactly. However when the rest of the solar system is added, then an exact solution is no longer possible. We can solve the movement numerically by essentially calculating where the planets will be after some time $t$. So based on where all the bodies are in the solar system we calculate a vector for earth and all of the other planets and advance the time by an hour. We repeat the calculation 525,960 times and we can predict where all the planets will be exactly one year from now. 

Now skipping several other mathematical models which offered better solutions than the planetary model, we arrive finally at the Schrödinger equation which gives quantum mechanical calculations for orbitals. The high level results are threefold. First the Schrödinger equation is subject to the same three body problem as before so an exact solution is only possible for a system with one electron and the nucleus. Second the electron orbitals are not discrete paths through 3D space but rather smeared out probability functions. Third different types of suborbitals (s,p,d and f) have different shapes. 

Now with the more realistic model for atomic structure we can again combine atomic orbitals to get molecular orbitals, with the new constraints introduced by quantum mechanics. But as we did before, when we create a molecular bonding orbital from two hydrogen atoms, then we get an antibonding orbital as well. Thus the same explanations used for hydrogen and helium bonding as before still apply, however the actual values of the bonding energies do change. 



I'd be remiss if I didn't point out something else. Going back to the the motions of the planets in the solar system we also stumble into something known as chaos theory in mathematics. Because of all the interactions small errors in the mass or velocities of the planets would result in much different results for the calculations. This is important in chemistry because the ethane molecules in a gas aren't a static entities, but rather dynamic ones. The molecules are rotating and the atoms are vibrating. Molecules are bumping into one another causing rotations and vibrations to change. 

Now trying to model all of this in one model is too complex for us to handle yet computationally. So chemists build models for different aspects of the problem and then try to piece together an overall picture. 

In the olden days, AAA would print a route map for someone wanting to drive from Augusta, Maine to Tampa, Florida. They didn't draw the whole route on a massive roll, but rather pieced the route together. Each page showed the pertinent decisions relative to that stretch of road. So a page in a large city might be a few miles, but out in the country a page might be a 100 miles. 

So rather like AAA, professional chemists don't have one massive bonding model that we keep hidden from new students. Rather we describe atomic orbitals and molecular bonding in a number of different ways depending on the situation. By piecing the overall model of a molecule together we can understand predict overall behavior.