you’ll notice as the minions take damage eventually they get a white life bar instead of the usual color meaning they will die the next time a champ hits them, however, when the minions hit them they do not die instantly. there is a purpose for this!
stage control is very important all game long.
most players from what i can see simply try to kill all the minions as fast as possible so they can team up with their own minions on the hero/tower they are in front of but this is actually very dangerous and unhealthy to you and your team because you get close to enemy tower at even levels and life it makes it way easier for the enemy to use their tower to kill you. true last hitting fixes this issue. if an enemy champ has hit the minions even once outside of true last hits(very last latest possible hit) enemy minions stack up on your side of the map causing the enemy champs to have to cross the middle into our territory in order to capture minion gold thus making it easier to gank them and harder to kill minions safely if they only get true last hits as well the minions stay at the middle of the map at very worst scenario you will not have to step into danger to get gold this way.
a true last hit should happen like this when the target minion has reached the point where one minions single attack will kill it you fire your shot after the minion swings but hit before it can connect with its attack what will happen is you will get the kill, your minions waste an entire attack that would have switched targets had u struck before the swing happened thus giving the enemy minions a life lead causing them to overextend slowly into your side of the map making it safer and safer to collect gold with out risk of enemy champs attacking you.
i hope this is helpful if you have any questions ideas or suggestions i am open to what you have to say and interested in your thoughts on how this effects gameplay.
all this being said i am not saying to always true last hit once you have enough of a lead you should start pushing up and making rotations to support the rest of the teams biddings and doings.