How does someone get teaching experience - i.e. seminars, lectures? It seems like current PhD students have a monopoly on having access to teaching positions at their universities, and that no one else on the outside can get these jobs.

That pretty much sums it up, yep!

Student often get put on zero hours contracts when they start their PhD. Try emailing the admin dept and seeing what they advise and whether they will take on external candidates under the same conditions (but I doubt it).

I take it you aren't a PhD student? If you want some teaching experience you could try colleges for positions like teaching assistant or support.

To lecture at university it usually requires a PhD depending on the subject. Hence why PhD students are usually given seminars/tutorials to teach, as it builds up their experience.