You can easily assign a claim to a member of your team by changing the Assigned To in the Claim Inspector on the right side within the claim.
First, open the claim you want to assign. The Assigned To field is located in the Claim Inspector and allows you to assign the claim to any active user.
You can see at-a-glance who is assigned to each claim, from the "Assigned To" column in the claims list screen.
It is also possible to filter on the Assigned To field to easily see a list of all claims assigned to a given team member.