If a gas expands, there is an increase in volume and therefore a positive $\Delta V$. However, because the gas is working on the surroundings when expanding into the surroundings, the process results in a loss of energy for the gas. Similarly if a gas contracts, the atmosphere is doing work on the gas, the system, as volume decreases. Therefore, there needs to be a negative sign as the system is the gas.
Work Step by Step
See explanation above.