The information for both fields usually either comes from text-mining the "Acknowledgements" or "Funding" section of a publication (if this content is available to us), or from structured metadata received from some of our publication data sources, e.g. Crossref or PubMed, which may contain the names / IDs of funders/grants.
Often, authors will cite the funder and / or ID/number of the grant that funded the research in the "Acknowledgements". If we can match the funder name to an organisation in our GRID database which has been labelled as a funder, the name of this funder will be listed in the "Funded by" section. Alternatively, the publisher may have provided funder names or IDs to Crossref or Pubmed, which we will then try and match to the funders in our database.
If, in addition, we have the funder's grants in Dimensions, and we can resolve a grant listed in the acknowledgements by its ID / project number to one of the funder's grants in Dimensions, this grant will be listed in the "Supporting grants" section.