In the year or so since this question first appeared on MO, there has been further progress on proving some of these identities, for example by Kadyrov and Mashurov and Dougherty-Bliss and Zeilberger.
The comments mention a scathing blog post by Persiflage, condemning the authors' self-promotion and failure to consult experts on continued fractions before claiming that their identities were new. I don't know if the blog post encouraged experts to get involved; possibly it had the opposite effect. At any rate, a number of the formulas have now either been identified as known or been proved, but many seem to be new and remain unproved (e.g., the two examples in the abstract, which have been changed since the original arXiv preprint was posted). IMO it would be good for more number theorists to get involved since the new formulas might be an indication of some new theory that remains to be discovered and elucidated.
As a side remark—a natural question to ask is whether any of these identities might lead to irrationality proofs. The only general theorem in this direction that I know about is a sufficient criterion due to Legendre (see Chrystal's Algebra, page 512, for a proof), and unfortunately none of the Ramanujan Machine's new identities seem to satisfy this criterion. Of course, maybe I overlooked something, or maybe they can be manipulated into a form that does satisfy the criterion.