Silumgar Spell-Eater, this effect only happens when the card is turned face up. So first it will have to be face down, usually from casting it face down for 3. Then it has to turn face up, usually by paying its megamorph cost of 4U. There are other ways of turning creatures face up such as Ixidor, Reality Sculptor, so the card doesn't care how it was turned face up. If cast normally it has no effect except if it is somehow turned face down with something like Ixidron.
The ability itself is a triggered ability with a target so the steps are as follows.
1. Opponent will cast a spell that you want countered
2. They pass priority hoping it will resolve, you take the special action of turning Silumgar Spell-Eater face up with megamorph.
3. Silumgar Spell-Eater's ability triggers and you choose your opponents spell as the target
4. Your opponent and you both pass priority without taking further action
5. Your opponent chooses whether or not to pay 3, if they don't the spell is countered and put into the graveyard. If they do then once again priority is passed around before the original spell resolves.
Silumgar Spell-Eater's last ability triggers only when it's turned face up, which normally happens only if you cast it face down with megamorph (by paying 3) (C.R. 702.36b), then pay its megamorph cost (4U) while it's on the battlefield face down under your control (C.R. 702.36e).
On the other hand, if you cast Silumgar Spell-Eater normally, without using megamorph, its last ability won't normally trigger. (See also C.R. 110.5, especially C.R. 110.5d.)
(Note that neither Silumgar Spell-Eater nor a face-down creature spell has flash, so you generally can't cast it in response to another spell anyway [C.R. 702.36a-b, 116.1a, 702.8a].)
EDIT (Jul. 19): Correctness edit.
EDIT (Jul. 9, 2019): Some rules were renumbered with Core Set 2020.
If I cast this normally, not metamorph, Is the counter spell ability still in play? For some reason I can't wrap my head around this one
The ability itself is a triggered ability with a target so the steps are as follows.
1. Opponent will cast a spell that you want countered
2. They pass priority hoping it will resolve, you take the special action of turning Silumgar Spell-Eater face up with megamorph.
3. Silumgar Spell-Eater's ability triggers and you choose your opponents spell as the target
4. Your opponent and you both pass priority without taking further action
5. Your opponent chooses whether or not to pay 3, if they don't the spell is countered and put into the graveyard. If they do then once again priority is passed around before the original spell resolves.
On the other hand, if you cast Silumgar Spell-Eater normally, without using megamorph, its last ability won't normally trigger. (See also C.R. 110.5, especially C.R. 110.5d.)
(Note that neither Silumgar Spell-Eater nor a face-down creature spell has flash, so you generally can't cast it in response to another spell anyway [C.R. 702.36a-b, 116.1a, 702.8a].)
EDIT (Jul. 19): Correctness edit.
EDIT (Jul. 9, 2019): Some rules were renumbered with Core Set 2020.