Within the framwork of the novel itself, the escape works quite simply by giving her freedom. It shows her personal desperation turned into an act of heroism and courage upon hearing the news that she is to be sold in order to settle debts.
On another level it works through the symbol of the river. One the one hand, the river marks the seperation between free and slave states ( the Ohio river to be precise) and thus symbolizes freedom and the geographical seperation between North and South, but on the other hand the river is also often used to indicate the increasing cruelty of slavery. "Being sold down the river" for instance is never a good thing, since the underlying idea is that slavery in Mississippi with its vast cotton plantations is much more cruel than in Kentucky.
We can also look to the original text, which is always a good idea when reading 19th century literature. The original was published as a series of installments in a magazine before ever appearing in book form, and in order to keep readers interested, there had to be a lot of cliffhangers or multiple highlights. That is why that particular type of texts appears to have such "racing plots" that make it sometimes hard to figure out where the climax is . I am not sure about the installment the included the escape scene, but I know that once the book was published in 1852, it included six illustrations ( also very common during this era) that served to highlight the most dramatic scenes in the plot. The part that was illustrated was the scene where Eliza tells Tom that they are about to be sold.