Quesillo is sweet, soft, unripened cow's milk cheese. This may not sound unusual, but the fact is that the huge U.S. dairy industry (with one minor exception -- read on) produces no product resembling this one. It's made from the fresh curd, and is neither aged nor salted (except lightly), similar in production to ricotta, farmer's, pot, Neufchâtel, Primost, Schmierkaese, Petit Gervais, bakers', and cottage cheese, but it is different from all of these.
One reason is freshness. In Chile, quesillo is frequently homemade from fresh cow's milk, and is highly perishable so is consumed in a few days. In the city, it can be purchased from the supermarket, but is still consumed when very fresh, and is not kept for more than 3 or 4 days, refrigerated. The U.S. dairy distribution and shopping habits are not compatible with this freshness requirement. Even cottage cheese contains salt, citric acid, sulfate, and phosphates to prevent ripening and the deterioration of flavor.
But it is the freshness and unprocessed purity of flavor which cannot be described that characterizes quesillo. It's "sweet" (as opposed to sour or salted), soft, and delicate, like sweet cream with texture. Any attempt to "bottle" this flavor results in a product which lacks the essential character of quesillo.
Of course, you can always make it at home. Rombauer & Becker ("The Joy of Cooking") used to give instructions, but nowadays they assume you have better things to do than go out an buy the stainless steel vessels, curd cutter, rennet, dairy thermometer, muslim sack, china cap, and other necessary tools and ingredients.
However, if you live in an area with both a large Mexican population (like San Diego), and a large dairy industry (like California), you have access to queso fresco, or fresh cheese, which is similar to quesillo in its fresh, mild flavor, but is crumbly in texture, and queso panela, which is silky like quesillo, but has a slightly riper flavor. Mexicans love unripened cheese, so it's very likely that a Mexican food market may carry some kind of queso fresco or panela. One popular type of fresh cheese is cotija, but this cheese is salted and semi-ripened to allow a longer shelf life, and many markets carry it for that reason. However, in San Diego, there are at least three large-volume Mexican supermarkets that sell genuine queso fresco in bulk (not pre-packaged or processed), and panela. (The company that makes it is Marquez Brothers, a Mexican company with a dairy in Southern California, and the brand is El Mexicano.)
Quesillo or queso fresco can be used in any dish calling for ricotta, cottage cheese, or feta. It is not a melting cheese, but is used to make Chilean cheesecake. It is usually served alone as a breakfast side, as a snack, or as an appetizer. It goes well in salads, and is a great substitute for mozarella in insalata caprese, the Italian tomato-cheese-basil-olive oil salad.