The phrase "to support this claim" means providing evidence, facts, or reasons to back up or prove a statement or argument that has been made. Full definition
I agree that the data we looked at this morning does not support any claim of a trend. (dotearth.blogs.nytimes.com)
They will be required to provide evidence supporting their claims for why the bail variation should be accepted. (mydefence.ca)
Frequent writing opportunities and opportunities to justify decisions and points of view can help students develop the skill of supporting claims with evidence. (apexlearning.com)