This is an example of an exceptionally long prepositional phrase that has been moved to the start of the sentence and joined with a comma. The sentence could also have been written:
On-site emergency response crews need to feel confident handling just about any type of crisis with remote Australian mine sites often long distances from professional emergency services.
I find it helpful to reduce sentences to their essential components and build them back up in order to understand their structure. In this case, it would go something like this:
Crews need to feel confident.
Crews need to feel confident handling [a] crisis.
Crews need to feel confident handling [a] crisis with remote sites [being] long distances from services.
With remote sites [being] long distances from services, crews need to feel confident handling a crisis.
The remaining words are all elaborating on the parts I included above: what kind of crews, what type of crisis, what kind of mine sites and how remote, etc.
The trickiest leap is from stage 2 to stage 3, mostly because the word "with" is used in an idiomatic way to mean something like "considering" or "given that".
Overall, this is an unwieldy sentence. Were I the editor for whoever wrote this, I might have suggested the following as an alternative:
Because professional emergency services are often far away, on-site emergency response crews at remote Australian mine sites must be prepared to handle any type of crisis.