He became frustrated with the inability of formal logic to explain everyday arguments, which prompted him to develop his own model of practical reasoning. The claim is the proposition or assertion an arguer wants another to accept.
Grounds refers to the proof or evidence an arguer offers.
Grounds can consist of statistics, quotations, reports, findings, physical evidence, or various forms of reasoning. The barometer is falling. The warrant is the inferential leap that connects the claim with the grounds. I'll bet she has an infection. It is a Golden Retriever.