Standard way of doing it:
- Outline what's wrong with the current undersized staff, where are the bottlenecks, what's being held up because there aren't enough people.
- Explain how this hurts the company's bottom line.
- Explain how hiring another person will solve the current problems, increase efficiency, and in the medium to long term, increase revenues more than the cost of hiring this new person.
If your case is well built, it'll be self-explanatory. If your boss/manager is reasonable, they will see the benefit of hiring a new person. If they don't seem to see the benefit and refuse to see the logic of your case, either
1/ you haven't built a good enough case (your fault)
2/ your boss is a jerk and you should quit
3/ something fishy is going on at your company (such as the company having run out of cash and being unable to hire, even if it'd make sense) and you probably should quit as well