In fact, it seems that people would drink both from the cup and from the saucer: After being served a hot beverage (e.g., coffee or tea) in a cup on top of a saucer, some would pour a small amount of the beverage onto the saucer, and then drink from the saucer. There's evidence that this was reasonably common into the 20th century, and evidence that some still do, for a variety of possible reasons! Though I've never seen this practice in person, the following is what I have aggregated from a bit of online searching.
A blog post contains some good links about the topic. I find the most compelling of them to be an article that discusses the use of deep saucers for this practice; the saucer is larger and deeper like a shallow bowl, alleviating some of the awkward logistical aspects of drinking from a flat or shallow saucer. Another anecdote, according to US government legend, George Washington (in a discussion with Thomas Jefferson) used the literal practice (cooling-off beverages in a saucer) as a metaphor for the cooling-off that occurs in the upper house of a bicameral legislature. Even if this exact exchange is embellished, it still suggests that the practice was common enough to be in the vernacular.
Fundamentally, pouring part of a hot beverage into a saucer would help cool it down faster. Many anecdotal sources have that as an assumed motivation; if you're in a hurry to start drinking your beverage (and are a bit careful about the pouring and the saucer-sipping) it's a practical way of doing so. Some other anecdotes suggest that it was fashionable or polite (maybe even fun?) -- perhaps like eating peas off the back of one's fork. Cooling a portion in the saucer would alleviate the need for potentially-impolite-seeming slurping or blowing across the beverage to cool it, substituting instead acrobatic-prowess-demonstrating yet possibly-more-polite sipping off of a saucer. Other discussion forums (one example) contain some additional anecdotes and pictures.