Usually, the only valid use case for the Pushgateway is for capturing the outcome of a service-level batch job. A "service-level" batch job is one which is not semantically related to a specific machine or job instance (for example, a batch job that deletes a number of users for an entire service).

How does Prometheus push gateway work?

The Prometheus Pushgateway exists to allow ephemeral and batch jobs to expose their metrics to Prometheus. So it's an intermediary service, where the clients (jobs) can push there metrics as desired before it shut down and Later Prometheus can Scrape those metrics from PushGateway as usual.

Is used to push logs into Prometheus?

How to feed logs into Prometheus? Short answer: Don't! Use something like the ELK stack instead. Longer answer: Prometheus is a system to collect and process metrics, not an event logging system.