Carreyrou originally got tipped off by pathologists saying that this was impossible, then they got sued into oblivion (see thepathologyblawg).
(IANAP but this is my understanding)
Some real tests rely on having a huge blood sample (e.g. 100ml), being filtered (e.g. by centrifuge) and then a test of a known sensitivity applied.
Theranos claimed that their tests were more sensitive so could work with smaller samples. Statistically this doesn't work because with a finger-prick test (e.g. 1ml) the sample is 100x less likely to contain the target cells - cells are integers.
Additionally finger-prick tests contain only capillary blood - they're filtered by the finger blood vessels only allowing tiny cells - some of the Theranos tests claimed they could detect markers that only exist in arterial and venous blood.