You can assign leads in two ways.
From the list of leads, click the arrow to pull up a list of users to assign to and click on the name of the user to assign to. You can scroll through to find more users.
If you are in the lead (access the full lead information from the list by clicking on the three dots), you will automatically be listed as the person it will assign to. This will only take effect if you action the lead. If you leave the lead without actioning it, it will remain in Unassigned.
To Change the assignee, click on your name and choose a different user