Exactly as the description says.
Okay let's say that my Eirika (-hp/+def) with 35 speed is going up against an unbuffed, neutral Lucina with 36 speed.
Normally neither unit would double, Eirika would deal 19 damage on Lucina while Lucina would deal 20 on Eirika. Eirika ends up with 19hp and Lucina with 24hp. Eirika actually comes out a little worse in this exchange.
Add Swordbreaker to the mix and suddenly Eirika gets two attacks for 19 damage each, a total of 38 damage. Lucina is knocked down to 5hp.
But wait, my Nino has Hone Attack 3. What happens in this situation when Eirika is boosted all the way up to 48 attack? Lucina is dead. Eirika lives.
In a situation when Eirika is going up against, say, a +spd Lyn with 40 speed she would normally get doubled. With Swordbreaker, Lyn will not double, but Eirika will, automatically. It overrides the speed stat when it comes to doubling against swords.