Just confused about how does the "Requires Proficiency" checkbox works when adding weapons or tools into a background, does it determines that your character needs to have proficiency on that item in order to use it or does it add the proficiency needed to use it? Also, leaving it unchecked turns that item usable by anyone, even if the character doesn't have proficiency with it?

That checkbox restricts the item(s) to only allow PCs proficient with them to take those pieces of equipment. If the PC is not proficient with that piece of equipment, the item will still be displayed in the character builder, but it won’t actually allow the player to select the item. The most common place to observe this behavior is not with a background, but with the cleric class in the builder where it restricts access to the warhammer to only PCs proficient with that weapon. In backgrounds it is most often used in association with backgrounds that also provide proficiency with the player’s choice of a musical instrument or gaming set. If the box is left unchecked, the attached equipment will be made available to all PCs, regardless of if they can use the equipment or not.

Although it is better practice to not restrict it. I see no reason to restrict players only to stuff they're proficient with because generally you can still use anything you're not proficient with - it's just less effective. Also, you may be wanting to gain proficiency later with a multiclass, subclass, feat or such as. The idea of restricting a starting option over proficiency is therefore rather stupid. 

I recommend not following WotC's stupidity on this.