How do I tell my employer I need more hours?

Tell your manager you need more hours and see what they say. If they won't give you more - yeah - get a second job.

It could have to do with you still being in training or it could be some other company policy.
Some companies have policies that part time workers only get a certain number of hours. Sometimes this is due to them paying benefits to works that are considered 'full time' or work a certain number of hours. In Ontario, Canada, full time is considered over 28 hours per week (at my previous job this was also the requirement for keeping additional benefits).
Speak to your boss about this. If it's training, you'll have to wait until you prove yourself. If it's company policy you'll have to work that out with them or get another job to make up the difference (keep in mind that due to scheduling, most employers are not keen on employees working another job, especially if scheduling isn't set and varies, it makes it more difficult for them and you'll often see fewer hours as a result)

There is no guarantee that there are more hours to give you - companies have a personnel budget and management has to do everything within that budget which limits overall hours. If you get more hours, someone else gets fewer hours. All you can do is mention to your manager that you are available. Also, let your co-workers know so that if someone needs coverage, they can come to you (note - this will be more ad hoc and outside your control - if you turn down too many people for these hours, they may stop asking).

Talk to them in private

Prove yourself to be an efficient worker and they will want you more.