Because дворе starts with two consonants in a row, в gets a vowel to make things easier to pronounce.
Yes, it's not a rule for every instance of two consonants in a row, it's only when (Russians) decide that adding в to the sound of the next two consonants would be difficult to pronounce.
But be aware of the fact that this rule is not always working.
For example, в здании, not во здании.
With a big non-native speaker caveat, the first sentence emphasizes that what the children are doing in the courtyard is playing there. The second I would translate as "The children are playing in the yard," and the emphasis is on where they're playing.
Russian tends to approach "yard" like they do arms and legs. A двор is just the spacial territory surrounding the house/building itself, without a distinction between front and back yard. You could say задний двор if talking about a back yard like what we have in the US.
Is there a major difference between сад and двор ? I know one means garden and the other courtyard but I'm not even sure if those are very different...
@Amawaku - There is a difference between the two (in both Russian and English) Сад is specifically a space meant for planting and cultivating plants (flowers and fruit/vegetable-bearing plants). It can be within a двор, which is just a yard of any kind - an open space associated with a house or building.
I can add that in Russian there is also "огоро́д" (kitchen-garden), it is a space on the ground meant to cultivating vegetables. The difference between "сад" and "огород" is: a "сад" has trees or/ with shrubs These trees and shrubs can are fruit/vegetable- bearing or not. But in "огород" every plant is vegetable-bearing, because people in Russia create "огород" to get their own organic vegetables.