On its own, this line seems pretty boring and unremarkable. Some girl named Julia is prancing around in a silk dress. Big whoop. But on closer look, this line actually tells us quite a bit about our speaker and his relationship to this Julia character. The very fact that he calls her "my Julia" suggests that he feels some affection for her—hey, maybe she's his main squeeze. Of course it could also suggest something a bit creepier, because that "my" conveys a sense of ownership.
Then, then, methinks, how sweetly flows That liquefaction of her clothes. (2-3)
The proximity of "sweetly" and "methinks" implies that the Speaker's own feelings are somehow contributing to his perception of Julia's clothing. It's all about how he sees her outfit, not what she's actually wearing. In fact, you could argue that Julia could be wearing a burlap sack for all we know, and that the speaker's love for her causes him to see it as something gorgeous.
O how that glittering taketh me! (6)
Swoon. Who doesn't love a little glitter and glam? Isn't the point of sequins to captivate the gawkers and oglers? Well it seems our speaker has totally fallen for Julia's fashion choices, and he's taken, swept up by, captivated by, her gorgeous appearance.