Lane priority is pushing out a lane faster than the opposing mid laner allowing time for roams, warding or invades.

If you can get kill pressure on the opposition than you can force a more passive playstyle and therefore they won’t be able to match your roam or be able to fight in jungle skirmishes.
More wave clear can push a wave a lot faster and force laners under tower. If they try to roam with you they sacrifice gold and experience from the minions.
Jungle pressure forces laners to playback and passive to avoid being ganked or even in certain cases being dived.
BUILD DEFENSIVE! Many people don't do this and it makes sense why. Because why build damage when I can kill someone in lane at half health. The thought goes both ways and therefore they are looking to kill you too and if you can survive the all-in you can ask for a gank or even retaliate later in the lane phase when you buy your offensive items. Little things like that can and do in facts make a difference in games being won or loss, and a comeback or a throw

Because lane priority gives you an advantage for roams it also translates to jungle priority which can be used to snowball you allied jungler and put the opposing jungler behind. With the jungle lead, it makes it easy to push the tempo to captures objectives such as drakes, herald, and tower plates if your lane has been doing really well maybe even first blood tower.
Lane priority can also put the enemy at a CS deficit meaning less gold for them to buy items and get stronger while you get ahead to buy more items and continue to oppress your lane. With the CS you can force the laner out of XP range of minions which by then your lane is won because they simply can't beat you 1v1 and is forced to call for help in which then your team can help and collapse or just take another objective off the map.

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