Net neutrality is about ensuring that your local internet Service Provider (ISP) treats all your internet traffic equally - not blocking or throttling it as they wish. Why would they want to do such thing? Mostly when they want to sell you a service and there is a free alternative out there, without net neutrality they can just block all access to it to users using their internet. It also enables them selling limited internet where you can only access certain things, which might not be terrible on individual level (as you chose to pay less) but is a bad thing on a bigger scale.
This particular vote only affects US residents and what dishonest practices their ISP can employ to get more cash out of them.
Because half the reason a lot of people cannot play HotS in the USA is because of so many people streaming that it is overloading some internet nodes causing them to drop packets. Under current regulations all packets have to be dropped and delayed equally, even if that packet is 1 of several megabytes worth per second used for a 4k HD stream that no one would care is dropped, or is a few kilobytes per second of game command packets that is extremely latency critical and dropping it will cause the player to experience lag due to retransmission. Technically under the new regulations internet companies could offer cheaper low priority bandwidth for streaming companies where latency is unimportant but quantity is while offering game servers more expensive high priority bandwidth where latency is extremely important but quantity is small.
Sounds good and sensible right? Well the problem, and reason why the previous regulation existed to stop this, is human greed. Instead of offering sensible QoS levels at sensible prices targeting different practical applications. They will use QoS to blackmail content providers into having to buy the most expensive "premium" QoS level to even stand a chance of having their packets delivered. Once everyone is subscribing to this premium QoS and the internet starts to behave like it does currently due to too much data and too little infrastructure they will then introduce a new "ultra premium" QoS level that is above "premium" with a larger cost. The black mail cycle will repeat with normal premium customers not being able to reliably deliver content unless they subscribe to the new ultra premium service. During this cycle no one bothers to invest in new infrastructure as you can just charge infinitely more for existing infrastructure by inventing new "extra ultra premium" QoS levels for it to sell.
Mostly when they want to sell you a service and there is a free alternative out there, without net neutrality they can just block all access to it to users using their internet. It also enables them selling limited internet where you can only access certain things, which might not be terrible on individual level (as you chose to pay less) but is a bad thing on a bigger scale.
The average private internet consumer will probably not notice the change at all, even to their internet bill, as there is no real way or point to meter them. It is the content providers who will be paying for the QoS, and that QoS can apply both ways (so they can pay for priority incoming and outgoing traffic). This is why Facebook, Google, etc are all against it since they will be the ones having to pay.
What the average private internet consumer will notice is that the QoS of the sites they visit and services they use will change, for better or worse. In the case of games, if the game company subscribes to a higher end QoS connection then their players might have a better play experience as connections that lagged before start to work with low ping. On the other hand if the game company goes with the cheapest QoS they can find the opposite can happen and where as before players could play with low ping suddenly the game becomes unplayable. This can only be made worse if huge bandwidth intensive content providers such as Netflix or Google use high end QoS to deliver their streaming services as although their streaming performance might improve slightly, it will totally trash peoples ability to play online games unless the companies use the same or better QoS in which case it becomes nothing better than it is now but costing more.
This is why "net neutrality" is important. This is why most sensible countries enforce it.