As gasoline - a refined product of oil - becomes cheaper, consumers will have more money in their pockets, which they will then go out and spend on other goods and services.
However, a research note from UBS indicates that the effect on the US economy might be a bit more limited than many expect for a couple reasons.
The bulk of gasoline spending is done by families at the top of the income distribution. Almost a third of gasoline spending comes from the top 20%, with another quarter coming from the next quintile:
UBS
While the top 20% are the largest consumers of gasoline, gas takes up a smaller share of those families' spending than those further down. People at the top of the income distribution are less likely than lower income families to spend their gas savings on other items, instead tucking that money away into their savings.
According to the authors of the UBS note, "These families' overall expenditures generally are not tightly constrained by income and credit availability. In other words, part of the lower gasoline expenditures likely will be reflected in a higher than otherwise overall personal saving rate."
The note points out a couple other effects of the drop in oil prices that might introduce some drag on growth. The drop in oil prices means that there will be less of a drive to build new wells and develop new fields in the US, leading to lower capital expenditures. Further, in the wake of falling oil prices, the dollar has exploded, which could lead to people buying more imports, moderately tempering GDP.
While the drop in oil prices should be an overall boon to the US economy, it's worth noting that there are other effects of that drop that could provide a small drag.