I think students can generally grasp multiplication by fractions, both in why the answers are reasonable and in applying real-world examples. Once students understand that multiplying by a number less than 1 gives a product that is smaller than the original number, remind students that division is the inverse of multiplication, and as such, dividing by a number less than 1 (but greater than zero) gives an answer that is larger than the dividend.
Real-world examples using length and time are easy to work with in a story problem. An example: I have 2 feet of ribbon and cut it into lengths of 1/2 foot each. How many pieces of ribbon do I have? Answer: 4.
Another example: Judy can make a friendship bracelet in 1/3 hour. She spends 3 hours making bracelets. How many bracelets does she make? Answer: 9.
This last example may have several ways to solve. Some would argue, therefore, it is not a good example. However, in the US, the CCSS allows for and encourages the development of multiple strategies in solving a problem. So, even though one student might think, "Judy makes three bracelets an hour" and multiply 3 times 3, this is mathematically equivalent to 3 divided by 1/3 -- we even teach kids to multiply by the reciprocal! I would suggest, then, that a problem like this be used during lecture time, allowing students to discuss the way they approached the problem.