The phrase often implies the speaker's empathy towards the overwhelming situation being described. The single mother groaned under the weight of working three jobs.
Pro Tip 2/3
Hyperbolic Nature
This expression can be used to exaggerate difficulties for dramatic effect. I'm groaning under the weight of so much homework!
Pro Tip 3/3
Metaphorical Use
Remember, it's usually not about physical weight but rather about stress or responsibility. She groaned under the weight of many roles at work.