the underlying data comes from the API via /v1/observations/popular_field_values.
for example, you could get data for Rudbeckia amplexicaulis (taxon 200073) from https://api.inaturalist.org/v1/observations/popular_field_values?taxon_id=200073&per_page=50&unannotated=true
…
you can also get the data via /v1/observations/histogram. this one will allow you to specify different time intervals (ex. month of year, monthly, daily, weekly, etc.), though you will have to make a separate request for each phenology value (flowering, budding, etc.).
for example, the corresponding month-of-year “flowering” data set for R. amplexicaulis could be obtained from https://api.inaturalist.org/v1/observations/histogram?term_id=12&term_value_id=13&taxon_id=200073&interval=month_of_year
this page may help to display this data in a more human-friendly format:
https://jumear.github.io/stirfry/iNatAPIv1_observation_histogram?term_id=12&term_value_id=13&taxon_id=200073&interval=month_of_year
annotation codes can be found here: https://forum.inaturalist.org/t/how-to-use-inaturalists-search-urls-wiki-part-2/18792#heading--annotations