It doesn't really. If you track your expenses consistently over the years (gnucash et.al.) you can trivially plot the increase in cost of living and the acceleration of that increase last handful of years.
They're also anonymous, so you should assume the worst. This is especially true when the supporting evidence is suspect (ie. a bunch of arbitrary picked numbers). On the other hand the BLS's methodology is documented (https://www.bls.gov/opub/hom/cpi/), so I'm willing to give them a pass even though their incentives might be skewed.
>(what would it be?)
their beliefs? "agenda" here doesn't have to be something nefarious. an anti-vaxxer's agenda is anti-vaccine, but that doesn't mean they're scheming to get everyone infected.