how did those nurses mix up muscle relaxer with an MMR vaccine?

Maybe similar containers and exhausted nurses?

After the second victim died nurse Luse Tauvale went to the rubbish to check the vaccine bottle and when finding it was Attracurium, she put it in a plastic bag and took it home.

I can’t imagine their pain:

The parents of both the infants broke down after the sentencing but told local reporters they have forgiven the two nurses.

https://www.1news.co.nz/2019/08/02/nurses-sentenced-to-five-years-in-prison-for-incorrectly-administering-mmr-vaccine-that-led-to-deaths-of-two-infants-in-samoa/

Do we have info about how this measles outbreak in Samoa started?

We have a reasonable guess:

New Zealand’s next outbreak would be the biggest yet.

Taking off in February, just 10 months ago, this would see the country shaken by a resurgence of such severity that, by the time of Samoa’s two-day lockdown, it would total more than 2,000 confirmed measles cases. Almost 80 per cent of those were in New Zealand’s biggest city, Auckland, where on the south side is a large Pacific Islander community.

And not only does this enjoy family links to Samoa, but daily flights to and from Apia. Last year, more than 80,000 such journeys were logged between the nations. And it is believed that from a passenger on one such trip, last August, the virus jumped from Auckland to Upolu.

https://www.telegraph.co.uk/news/measles-in-samoa/