Why do we use the electron volt?

The electron-volt is a convenient unit of energy when considering electrons moving between points at different potentials. The convenience came from having numerical values which are around or greater than one, $1 \rm eV = 1.6 \times 10^{-19} \rm J$. It was first used in the 1930s.

So one perhaps has a better "feel" for the difference between 1 and 100 eV than $1.6 \times 10^{-19} \rm J$ and $1.6 \times 10^{-17} \rm J$ and the value in electron volts is easier to write.
Electron energy levels are conveniently quoted in electron-volts and then nuclear energy levels in MeV show a clear difference in terms of scale.

Then using eV/c² with the appropriate prefix as a unit of mass also becomes convenient; e.g. the mass of the electron as 500 keV/c² and that of the proton as 1 GeV/c².

It is not an SI unit but is retained because as well as being convenient it was and still is in widespread use in the scientific community.

Addressing only why it is used/useful in science today, not why or how it came about

The other answers seem to come from a particle physicist's point of view; for a chemist the electronvolt is convenient as well:

• 1-10 eV: energy required to break a typical chemical bond
• 5-25 eV: energy required to ionize (remove an electron from) a neutral atom
• 1-5 eV: energy of photons of visible light

Please note these are "order of magnitude" ranges.

"Historically, the electronvolt was devised as a standard unit of measure through its usefulness in electrostatic particle accelerator sciences because a particle with charge $q$ has an energy $E = qV$ after passing through the potential $V$; if $q$ is quoted in integer units of the elementary charge and the terminal bias in volts, one gets an energy in eV."
source

Further, you will have to admit that energies written as $x \cdot 10^{-19} \textrm{J}$ are not the most useful numbers to work with. Using an arbitrary standard quantity (e.g. Angstrom) as comparison to obtain numbers that do actually mean something to us and that make it easier for us to talk about them is quite a widespread custom in physics.