The word push comes from English and literally means to press/push/drive. That's exactly what a push message does: it proactively informs you about events; it "pushes" you a message.

In the smart home area, push notifications are of great importance: Smart home devices that are connected to your cell phone automatically send you messages with the desired information. The essentials smart home surveillance camera sends you a message when movements are detected in the monitored area. And the it Smart Cooking hot air fryer 5.5l sends you a message when the cooking process is complete.