I work on the coalface of covid response -- PP responses are good.
I wouldn't say complete mystery, but there's always some uncertainty. All of the high quality evidence points to covid being mostly (probably overwhelmingly) transmitted by direct contact with droplets. Contagious person needs to do something to spit those droplets out or it's very unlikely to transmit.
Do these people go on to develop symptoms - so they are in the pre-symptomatic phase?
Most often yes one study I'm involved in, those without symptoms who had a positive test majority got symptoms within 2 weeks. The others remained truly asymptomatic. Other studies have similar findings.
Or is the test picking up remnants of an old infection from when they did have symptoms? Or are they false positives?
Test picks up matching RNA, but the RNA may not be viable (so unable to infect anyone). So post-symptomatic positives are very possible. % of false positives is probably small. There are a few different PCR tests on the market, I'm not sure how reliable each one is.
Or do people truly have this virus to detectable levels and not get poorly? Why ?
Some people can have high viral load & not be sick: nobody knows why. Also The tests are very sensitive is why they can detect levels that are so low that they can't make a person feel ill. We don't know why some immune systems stop virus from making people feel ill. It's normal with most infections that some people get it bad & others barely at all. Zika & polio are good examples: symptoms can range from nothing to very bad or even lethal.
The second part of the question is, if people can have covid-19 and never experience symptoms, is there research yet on if they are contagious?
Hot topic in active research! Likely to be low-to very low risk for infecting others, for reasons others said. But low risk is not Zero risk.