He can choose an appetizer any of 3 ways.

For each of those 3 ways that he can choose an appetizer,
he can choose the main course any of 4 ways.

That's 3×4 or 12 ways that he can choose an appetizer and a main course.

For each of those 3×4 or 12 ways that he can choose an appetizer and a
main dish, he can choose a dessert any of 2 ways.

That's 3×4×2 or 24 ways that he can choose an appetizer, a main course,
and a dessert.

Answer: 3×4×2 = 24 ways

Why does it work to multiply the number of choices of each?

Let the Appetizers be A1, A2, and A3
Let the main courses be M1, M2, M3, and M4
Let the desserts be D1 and D2

Then the 24 meals are:

 1. A1, M1, D1
 2. A1, M1, D2
 3. A1, M2, D1
 4. A1, M2, D2
 5. A1, M3, D1
 6. A1, M3, D2
 7. A1, M4, D1
 8. A1, M4, D2
 9. A2, M1, D1
10. A2, M1, D2
11. A2, M2, D1
12. A2, M2, D2
13. A2, M3, D1
14. A2, M3, D2
15. A2, M4, D1
16. A2, M4, D2
17. A3, M1, D1
18. A3, M1, D2
19. A3, M2, D1
20. A3, M2, D2
21. A3, M3, D1
22. A3, M3, D2
23. A3, M4, D1
24. A3, M4, D2

That's why it works to multiply the number of choices for each.

Edwin