How can you hope to clearly explain something you yourself do not understand well?
I have found the following line of reasoning is quite convincing for many skeptics:
How hard did precalculus seem when you took it? After taking a semester of calculus, how did your perception change? How about after taking 3 semesters of calculus?
Mathematics is cumulative. The more you learn, the more you reinforce the basics.
By the time you finish the standard sequence of several semesters of calculus, differential equations and linear algebra, you start to find that algebra and the basic precalculus material is equally difficult with basic arithmetic. When first confronted with derivatives, maybe they seemed weird/difficult/tricky, but after a couple of semesters of material building on the basics, taking a derivative is like adding 2 and 2 or breathing.
The point is, the more you know, the more you will consider basic/easy/trivial. When teaching a subject, you want to be able to focus on how you are presenting the material - how to best communicate this to you students. You don't want to be fighting with understanding it yourself.
Along the same lines, the more you know, the better perspective you have as to why different topics are important. The more math you learn, the better you will be armed when confronted with a skeptical student of your own who asks, "Why do I need to learn this?"
Admittedly modern algebra as taught at most universities is a hard sell for you typical unmotivated future highschool teacher (why you would enter such a profession half-hearted, I do not know, but it seems common). If you focus on ring theory, you have a chance at selling the connections to factorization and the like. Groups are trickier because, honestly, that specific material may very well be completely unapplicable. However, students who spend a semester really engaging their modern algebra (or any other proof oriented) class should leave with at the very least bullet proof logic skills. This of course is of immeasurable value when teaching mathematics.
Anyway, that's how I've approached this topic with many students (and been successful as far as I can tell).