For Shaco, good champions to use are
- Lee Sin

- Udyr

- Shyvana

- Rek'Sai

I suppose the reason the site listed Rek'Sai is because she cans till "sense" Shaco even if he is in stealth. Hence, she makes a good counter jungler. As for Lee Sin, he is generally good for playing against stealth champions as he has abilities that reveal them.
General Counter Tips:
Always keep an eye out for a puff of smoke in the fog of war - this means Shaco is nearby.
If Shaco uses his Deceive to enter fights, he won't be able to use it again for a while. Try turning and focusing him if possible.
Counterjungling Shaco's red buff or blue buff can greatly limit his early gank potential.
It's good to ward your jungle (especially early game) as he can steal your buff and get away.
Vision wards are effective at knowing where Shaco is during ganks.
Shaco can be a frustrating champion to kill due to the quick and easy escapes with Deceive, bringing him down with a disable or a fast burst will make him a much easier prey.