Because she recognized them, he was able to realize the threat Egwene was facing, and that she was wrong (which no one should really need any evidence to know, but this IS Gawyn after all) in her assumption that she was only being stalked by Mesaana & the Black Ajah or that she had nothing to fear from any non-channeler. As a result of his realizing that, yes, there WERE assassins stalking her, and they WERE capable of getting to a channeler, especially one arrogant enough to demand no guards because she's just the bestest and specialest Aes Sedai ever, and will easily trap anyone dumb enough to try to assassinate her, he was motivated to return in time to save her so she could survive to keep being petulant and obstructionist in just the right way as to make Rand's ultimate plan come to fruition. If she hadn't, the Tower might have chosen a replacement who might have been inclined to cooperate or help Rand somehow, and there is just no way the heroes could wrap their brains around such a game changer fast enough to adjust and deal with the new situation. Rand has obviously made some careful plans, and possibly even some desperate gambles, all predicated on the assumption of Egwene's character, namely her arrogance, intransigence and reflexive opposition to anything he tries. If Gawyn fails to save her life, those plans all come to ruin, and thanks to her attitude and obnoxious way of dealing with him, he would never have been motivated to return in time unless he knew she was in a unique danger from which on he could save her. He would never realized that threat existed without the Seanchan woman recognizing the ring he picked up. That's really all there is to it.