The testicular veins are responsible for draining deoxygenated blood from the testes to the inferior vena cava or left renal vein, with the right testicular vein directly connecting to the inferior vena cava and the left testicular vein draining into the left renal vein. This venous drainage system is crucial for maintaining optimal testicular temperature and function, and any obstruction or dysfunction can lead to conditions such as varicocele.