A use-by date is about safety — do not eat the food after it. A best-before date is about quality: the food may be past its best but is usually still safe if stored properly.
For eggs, a simple freshness check is to place one in water: very fresh eggs sink and lie flat, while older ones stand up or float.
The takeaway
Respect use-by dates for safety, but use judgement and simple checks around best-before dates to cut waste.