802.11h opens up a bunch more channels in the 5GHz band for you to use with 802.11a/n/ac.
Without H, you can only use channels 36-48, and maybe 149-165 if they are allowed in your region. So a max of 9 channels.
With H, you can use 52-64 and 100-144 (again, there's some variation by region) for a max of 16 additional channels.
Said another way, if you don't enable H, you may not be able to use a ton of access points around you. You may completely fail to see some networks, and you may find slow spots or dead spots in the coverage of large enterprise, institutional, or public access networks, because the nearest APs that your client should have found (and roamed to) happened to be using channels that require 802.11h. Or your client may have to fall back to connecting to APs in the slow/congested 2.4GHz band.