It's less important to go over the history of the orcs in terms of the wars of Azeroth. We've done it, many times. What's interesting to discuss is the orcish acclimation to Azeroth, and furthermore, Azeroth's acclimation to the orcs.
The orcs have changed during their time on Azeroth from a nation of blood-drunk servants of evil to a people leading a faction that seeks global dominance in the name of a legacy they've invented for themselves. Orcs today have a warrior culture that comprises elements from Blackhand's Horde, their past on Draenor, and a great deal derived from Thrall's efforts to create unity and give his people a culture again. While the modern orcish nation is led by Garrosh Hellscream, a brown Mag'har orc, it cannot be said that most orcs of the Horde really understand Draenor. The Second War ended more than 20 years ago, and many of the orcs of today are the children of those who fought in it.
This must be understood: Many orcs alive today on Azeroth have never even seen Draenor. Those who did last saw it 20 years ago. Azeroth is their home as far as they're concerned, either the only home they've ever known or the one they've known for decades. From the perspective of most orcs, Draenor is effectively gone. Oh, many of them are aware that Outland exists, and there are those orcs who have been there in recent years, but most orcs living today have never seen it at worst and saw it decades ago at best. To them, Draenor is nearly a myth, and Garrosh Hellscream becomes a mythic figure as a actual brown orc, an uncorrupted Mag'har who lived most of his life on that long-lost homeworld. It is this, as much as his lineage as the son of Hellscream, that has made him a legend among the orcs of the Horde today.