There are a lot of directions you can go when portraying a Bard, especially given that the class is typically depicted as a jack-of-all-trades. In 13th Age, it's arguably the most complex class in the core book so there are a lot of ways you can customize your Bard mechanically, but Flexible Attacks are a surprisingly open-ended springboard for narratively differentiating your Bard's fighting style.
The most obvious interpretation to take is that Battle Cries are exactly that - your Bard shouts commands, and your allies follow them. The mechanical benefits represent the effectiveness of your commands. While this is certainly a valid take on these Flexible Attacks, today I was trying to think about ways to step on the toes of the upcoming Commander class less. Because ultimately, that's the Commander's main schtick, and in that sense it's a bit unfortunate that the default flavor for the Bard's attacks overlaps with it so much.
Another option is to play off the whole magic-from-music angle for as much as it's worth. Your Bard isn't shouting orders when he uses his battle cries, but singing, and the effects of the attacks are a direct result of the verses recited.
Some players don't like the "guy who sings when he's fighting" archetype, and admittedly in some settings and groups it can be a little out of place. It's no big leap to say that your Bard casts minor magics while fighting. "Pull It Together" isn't so different from the Cleric's "Heal" spell, though it's quicker, if less reliable. Maybe the reason why it's less reliable is that the somatic components of the spell are the jabs and slashes of your blade. If your enemy parries your sword at the wrong time, the spell fizzles out. Or maybe the blade itself holds the magic, and it's not released unless you strike properly with the weapon. Any of these explanations and more are perfectly well-supported by the flexible attack mechanics (you gain some benefit depending on the results of your melee attacks). You're a "gish," much like the Swordmage, Bladesinger, Hexblade, etc.
One particularly intriguing interpretation is that Battle Cries are not magical, but they're not literal shouts either. This type of Bard is a clever, flashy swordsman, adept at feinting, misdirection, and providing openings for allies. This could be your swashbuckler, or you could inject the performance elements back into your description but instead of song, go with dance. This is most appropriate for a Dex-based Bard (and most are anyways), and I like to visualize this fighting style as being like River Tam's fight scene from the end of Serenity. You use a dancer's grace to keep a foe busy enough to give an ally a breather or an opening, inspire allies with the sheer beauty of your movements, cause enemies to over-extend as they try to match your strokes with their substantially less flexible bodies, etc.
The point is, remember not to box yourself in when you create your character and see "Battle Cries" written in the book. During play, embellish the descriptions of your Flexible Attacks to make it clear to the other players what's going on. You don't have to do this every time you roll the d20, obviously (that would grow tiresome), but you should definitely embrace the fact that 13th Age is a narrative game, and note that the rules for Battle Cries leave them very open-ended (except for the name...).