I'm an engineer who designs solar array for a living, here's how the math breaks down in fairly typical round numbers.

The all-in cost is around $2-3k per kilowatt (thats equipment, installation, permitting, utility approvals, etc), so a 5kW system (pretty typical residential size) would cost $10-15k. Each kilowatt produces about 1000-1500 kWh every year (depending on your latitude and how much sun your roof gets), so if your electric company charges you $0.10 per kWh, that 5kW system will generate $500-750 worth of energy annually. Without incentives it would pay itself off in 20 or 30 years, but if your state has good solar incentives that can be much shorter, if you pay a lot more for electricity it pays itself off sooner as well.