How can you demonstrate to employers that you have effective writing and speaking skills?

Ironically, you have gone part of the way to answering your own question.

I suspect that what you really mean is how can you prove, rather than demonstrate, since by demonstrating your effectiveness at these two key skills, you will achieve your objective.

A good way of proving key skills is to use them at every opportunity, for instance, when you give your employer a verbal message, back it up with a written summary.

Try to develop the good habit of looking for feedback relating to your communication: is it as effective as you intend it to be? Is it ever ambiguous? Is it clear? Have you effectively summarised?

Bear in mind that good communication skills need to be practised.

The most important thing is ALWAYS engage brain before opening mouth.