(Note for advanced readers: this is meant to be an entry-level introduction, and so there will necessarily be some oversimplifications; when in doubt, consult a reputable textbook.)
what is quark-gluon plasma?
Ordinarily, at temperatures below a few trillion Kelvin, the atomic nucleus is a collection of protons and neutrons. Each proton and neutron, in turn, is made of a triplet* of fundamental particles called quarks. A proton is different from a neutron because it's made of a different triplet of quarks. The quarks are bound extremely tightly into these triplets by the strong nuclear force, and the carrier of the strong nuclear force is the gluon, which you can think of as constantly being exchanged between the quarks in a proton or neutron. In short, quarks are bound together into triplets by gluons.
The fact that quarks tend to associate into triplets like this is no accident - it stems from the fact that the strong nuclear force has three different kinds of charge (called color charge in a nod to the three primary colors). The strong interaction has an interesting feature called color confinement - basically, objects with a nonzero color charge cannot be isolated. If you try to pull two color-charged objects apart, you'll spend so much energy doing so that you will actually create new oppositely-color-charged matter, leaving you with two non-color-charged ("colorless") objects in the end. So everything made of quarks must be colorless. There are generally two** ways to achieve this: either with a quark of one color and an antiquark of the corresponding anticolor, or with three quarks, each of a different color, so quarks are generally always found in pairs or triplets. Protons and neutrons are of the latter type. Note that we don't completely understand color confinement, and studying it quantitatively is an active area of current research.
The above applies for any matter whose temperature is below a few trillion Kelvin. If the temperature is above a few trillion Kelvin, though, the nature of color confinement apparently changes. Instead of being bound into triplets, the quarks and gluons all interact with each other at once and begin to flow, in a single blob of quantum fluid. This new fluid-like nuclear state of matter, in which quarks and gluons are freed from their usual arrangements, is called quark-gluon plasma. It's called a "plasma" because the behavior here is analogous to the formation of an ordinary plasma, in which electrons and ions have enough energy to separate from each other.
how can extreme cold and high energy exist together?
In the specific instance you're quoting here, they don't. Bose-Einstein condensates exist at extremely cold temperatures, and quark-gluon plasma exists at extremely high energies.
