Insurance will pay for it, right?

The term dental insurance is a bit misleading.  We're used to insurance that will pay if something happens and usually has big limits.  Dental Insurance is expected to be used and has a low limit, usually on a yearly basis (typically $1000-1500 per year).
The problem is your dental issues often don't try to match your limits. You either don't have any dental problems so you pay more in premiums than you use (but your limit starts over the next year) or you break something and it exceeds your limits. Dental insurance helps but it seldom covers everything.
If your employer has a health savings account (HSA), put money in that each year and if you don't use it, it carries over to the next year so you don't lose benefits.

Why does a toothache hurt so much?

A tooth is made up of three parts. The outer layer is enamel, the bulk of the tooth is dentin and the center has what's called pulp tissue.  The pulp contains nerve tissue and blood vessels to bring nourishment to the tooth.  The pulp is what most people call "the nerve" because that's what transmits feeling and pain.
When a cavity gets deep, the bacteria can reach the pulp.  Irritation and sometimes infection occurs and the result is pain.  It's the body's way of telling your tooth you waited too long to fix it's problem.
Inflammation and pressure down the root occurs and deep throbbing pain that is hard to stop.  Usually root canal treatment (to remove the pulp) or extraction (to remove the tooth) is necessary.  Your dentist can help you determine which is best for you.