Why does this toilet fill valve leak around the top?
That's a fairly popular fill valve. To open it:


Shutoff the water, you don't want a geyser. 
Lift the float to the top position.
While holding up the float, push down on the top cap and twist the cap a quarter turn (I believe counter clockwise). 


The cap will then lift off and you can inspect and clean any parts. Before reinstalling the cap, move any small parts to a safe location, get a small cup to place over the top of the valve, and turn on the water supply. This will jet water straight up from the top of the valve and clear out any debris.

Reinstalling the cap is the reverse of the above procedure. Put the parts back, position the cap back on top, lift the float, press down on the cap, and twist.

Note that if cleaning it doesn't work, it's fairly simple to replace this fill valve.