The clearance setting (of a zone) determines the distance between copper features of different nets. (example between gnd and any of your signals)
The clearance you can use is determined by your manufacturers capabilities and by the voltage you board is used for.
In the picture you posted there is quite a large distance between the zone and the tracks on your board.
You can change the clearance settings of your zone by pressing e while your mouse hovers above the outline of the zone. (On the left side of the properties window there are 2 settings. Clearance and minimum width.)
Normally you can set this stuff equal to what you use for everything else.
Again, check what your manufacturer is capable of. But i'm sure the standard settings for zones is much larger than what any normal fab can do.
(Zones start with approximately double to what the standard drc settings are.)
As far as i remember this tutorial explains these stuff quite well:
(Clearances are explained on page 9)