I have a spending journal structured by day. Included in the columns are
Amount Spent and Category (worksheet 1).

In preparation for creating a pie chart, I need to sum each category
(worksheet 2).

I'm looking for a formula that will look at the category and, if it matches
the criteria (i.e., Eat Out), will include the amount spent in the sum.

Try this:
SUMIF(range,criteria,sum_range)

Example:
=SUMIF(A1:A15,"Eat Out",B1:B15)