See: Downloading and licensing ESXi 5.x (2014574) @vmware.com, basically you fill out a registration form during ESXi download which result in an activation email, and if you follow through with the activation you will receive a license key for the free ESXi version. Once you have the free license key you add it and then switch your host over to it.
With regards to Veeam, it no longer works with the free ESXi at all, see: Veeam and Free ESXi – it’s official now @veeam.com (from 2009!) which states:
Recently, VMware requested that Veeam Software discontinue support for
ESXi Free in Veeam Backup and Replication in order to comply with
VMware’s updated licensing policy. In light of VMware's request, and
our close technical partnership, Veeam Backup and Replication will no
longer support ESXi Free. We will still continue to offer support for
ESXi Free to existing Veeam customers who purchased Backup &
Replication prior to version 3.1.