How does bombarding a molecule with electrons cause ionization?
This type of ionization is known as electron ionization (there are many types of ionization methods used for MS). One possible reaction for EI is:
$$\ce{M +e- -&gt;M^{•+} +2e-}$$ source

If the incoming electron has the correct energy to interact and passes close enough to the molecule, it's electric field can impart enough energy to one of the molecule's electrons for it to be ejected. What happens to the electrons after that isn't particularly important as far as ionization is concerned, but in most EI sources, electrons are emitted perpendicular to the axis of the sample source and are collected on the other side by an electrode, preventing charging of the source chamber and focusing the emitted electrons into a beam.