Why do negative ions flow into a cell in an inhibitory synapse, even though a neuron has a negative resting potential?
Short answer
Negative ions can flow against the electrical gradient into the cell, provided their concentration gradient across the cell membrane is large enough.  

Background
When an ion channel opens, the resultant ion flow is dependent on two things; the membrane potential (which is indeed negative at rest) and the concentration gradient of the ion.

Consider the activation of a GABAA receptor. GABA is the principial inhibitory neurotransmitter in the nervous system. GABA opens the associated Cl- channel on GABAA receptors. 

Using the Nernst equation one can approximate the membrane potential (Veq) at which an ion is in equilibrium. It mainly depends on the concentration of the ion X outside and inside the cell ([X]o and [X]i, respectively).[Cl]o is approximately 103 mM, and [Cl]i 4 mM. Hence, there exists a large concentration gradient with lots of it outside the cell.


Nernst equation. Source: Physiology Web

In a neuron, the Nernst potential of Cl- is around -71 mV. Hence, given that the resting membrane potential is approximately -70 mV for a typical neuron, no Cl- flux will occur when Cl- channels open, because the resting membrane potential is equal to the equilibrium potential of Cl-. However, when a neuron is depolarized by stimulatory neurotransmitters (e.g., glutamate), the membrane potential may be reduced to, say, -60 mV. This is still more negative than the action potential threshold (about -55 mV), but very close to it. So without any inhibition the neuron would be close to firing a spike. However, when GABA is released presynaptically, opening of the Cl- channels coupled to GABAA receptors will result in Cl- influx, because the membrane potential is higher (more positive) than the equilibrium potential of Cl-. In effect, Cl- will enter the cell until its equilibrium potential is reached, re-polarizing the neuron back to -70 mV and hence inhibiting it.