Slow-pushing is a way to be in two places at once.

If your minion waves are equal, you can start a slow-push by killing off 2-4 minions in the enemy wave. The more you kill the faster the wave pushes.

You can kill off the entire enemy minion wave if your minion wave is already missing some minions.

You generally start a slow-push in the mid to late game when your team wants to go for an objective.

By building up a large concentration of minions in your lane, you don't have to be there to push out enemy minion waves, your minions will do that for you. This gives you the opportunity to leave your lane to go help take the objective your team is interested in. If the giant minion wave that builds up starts to threaten a tower or an inhibitor, at least one champion on the enemy will likely be sent to deal with it. This means that your team can go after the objective with a champion advantage.

You can also set up a slow push when you want to roam to another lane if you think you will be gone from your lane for a while. If you just plan a quick in and out, then just shove the enemy minion wave into the turret without bothering to set up a slow push.

Professional teams use this tactic a lot. And the better ones will prioritize minion wave control and set up either all lanes to push or slow-push or set up a slow-push in a key lane, usually cross-map, when they want to contest a particular objective. E.g. setting up a slow-push in bot lane when they want to take Baron.

Minion wave control is why you don't see good teams running back to deal with a minion wave heading for their outer Bot base turret when they are trying to take an objective on the top side of the map.