Electron degeneracy pressure arises from quantum effects. It follows directly from the Pauli exclusion principle, which states the following: two identical fermions cannot occupy the same quantum state (a set of values that describes the state of a particle) simultaneously. Now, an electron is a fermion and thus this principle applies to it. An intuitive way of understanding this principle is to say that two electrons cannot occupy the same position simultaneously - this fact is important to get an intuitive idea of where the "pressure" comes from.
Now, to understand what the electron degeneracy pressure is, let us imagine a bunch of electrons inside a box that can change its volume as we wish. Then let's make the box smaller, squeezing the electrons inside.
As these electrons get squeezed together, their quantum states start to look the same. To avoid becoming equal, as the Pauli exclusion principle prohibits, they are required to have different energy levels (or, to become more energized). So, if you wish to add another electron to this box, you need to raise the energy level of another electron inside the box (thus, changing its quantum state). It's this energy that is required to add another electron to the box that is perceived as a "pressure."
Now, keep in mind this is a superficial explanation of how this pressure works, and there is a lot of math behind it.
I hope I gave you some enlightment on the subject!