Carbon fiber is… hard to explain. Lets start with 'why is it so strong'.
Lets start at the beginning and quickly run through. Lets start with the table of elements. These elements all have different numbers of electrons/protons/neutrons. A lot of elements 'do not like' sitting around with uneven electrons (too many or too few). Some 'want' to gain and electron, and other to lose an electron. A common example is table salt (NaCl). The Na has an extra electron it wants to lose and and the Cl wants another electron. The Na loses an electron and becomes a positive ion (Na+) and the Cl gains an electron (becoming Cl-). Positive and negative charges attract and electrostatic forces hold these two atoms together in whats called an ionic bond. However when Cl meets something that isn't offering its electron so freely, frequently they wind up 'sharing' and forming covalent bonds. These are two intramolecular forces.
There are some other forces but this is the gist of what you need to know. These bonds hold together the atoms in a single molecule, but what about holding together the molecules in a substance. In a gas each molecule is free and floating around on its own etc, while in a solid the molecules are all held together by intermolecular forces.
In essence each molecule pulls on nearby molecules (by one of the above methods) and they are all held together. The properties of the molecules in a substance define how strongly they will attract eachother and how hard it is to 'seperate them'.
There is a distinct disparity between intramolecular forces and intermolecular forces. Intramolecular forces are much much stronger (dozens of times, sometimes hundreds of times) stronger. This means as a rule of thumb it is much easier to seperate two molecules bound with intermolecular forces than it is to seperate two atoms within a single stable molecule.
If you want to say cut a block of ice in two the force you have to overcome is the combined intermolecular force (the water molecules pulling eachother together) along the line you wish to cut it.
Now we can start to look at carbon. The reason diamond is the strongest material on earth is that it forms (simplistically) one giant molecule. This means if you want to say cut a diamond down the middle you must break all the covalent bonds of all the atoms in the line you wish to cut. These covalent bonds are so much stronger than the inermolecular forces of the ice block that the energy required is much much greater and diamond is as a result incredibly hard.
Carbon fiber is similar to diamond, but instead of the entire fiber being one giant molecule, it is a collection of smaller (but still very large) molecules. The basic intermolecular forces still apply, but the each of these huge carbon molecules (relatively) is large enough to interlock with other molecules in a totally different way. It forms large sheets and the sheets of these molecules fold together and for a physical 'interlock' that makes them significantly harder to pull apart.
The result is not as strong as diamond, but also MUCH MUCH cheaper to produce, and can be more easily worked and shaped. It still takes a very large amount of energy to break carbon fiber (relative to its weight). One primary downside is that the interlocking makes it brittle. I can hold large strains relatively unchanged until you have enough force to 'break the lock' and then it snaps.
So in summary carbon fiber is a specially produced form of carbon that has large interlocked molecules that give is a very high strength to weight ratio. They normally mix it with a resin of some kind to help hold all those interlocked things in shape and assist in fitting it to moulds etc (also carbon fibers damage cells, so you do not really want to be touching them directly). The result could be compared to reinforced concrete, with the resin being the concrete and the carbon fibers being the steel reinforcement.
Hope this helps.