Multiplex PCR question

The key distinction of multiplex PCR is just that you amplify multiple independent targets from within your larger DNA template, through use of multiple sets of PCR primers.

In any kind of PCR, there are multiple possible uses and consenquently many different ways of getting a read-out. You could determine the size of the product on a gel, obtain the DNA sequence, quantitate using qPCR etc.
However, when you have multiple targets in the mix, you need to find a way to make sure that you can get a separate read-out from each, so that the signals from multiple targets don't pile up on top of each other in an unintelligible mess.

E.g. if your assay is to detect subtle differences between individuals in the size of a microsatellite for DNA fingerprinting, you could multiplex multiple microsatellites so long as each has a different size range and there is no overlap between them.

A quick look at the video you linked shows a multiplex qPCR method for detecting and potentially quantitating abundance of pathogens.
The assay seems to work by ensuring that each pathogen yields a product with a different melting temperature, meaning that each generates a separate discrete peak in the melting curve read-out.

Nowadays multiplex PCR is increasingly being used in conjunction with next generation sequencing for high throughput genotyping of tens to hundreds of genetic markers in hundreds or thousands of individuals.
Multiple genetic loci can be amplified at once, and then millions of individual DNA molecules from the resulting mixture can be isolated and sequenced, allowing you to identify the locus from the sequence and additionally to detect any variation within it that can be used to characterise an individual.

Problems with multiplexing include cross-hybridisation between primers or their amplicons, resulting in all sorts of artefact sequences that reduce your ability to make sense of the output. Careful primer design and pilot experiments are needed to get round this.