It is an interjection (感動詞), as indicated in the dictionary definition.
So you can use it as such. You could say just しめしめ、、、 for example, with the "remaining" part (explaining the interjection) implied. As you could say just やれやれ for example.
But as you noticed, it is more something one says to himself (even if it is voiced), so you wouldn't express it in the same way as やった！ for example, which you could also shout to the whole world to hear. A bit like a chuckle is not really for others to hear for example. You would typically picture someone using this interjection as grinning.
There's more details about this part of the definition here:
しめしめは人に聞こえないような声や心の中で呟くといったように密かに喜ぶ際に使われる。
On the meaning itself:
しめしめ has been abbreviated from 占めた which is can be used exactly the same (as an interjection) and has the same meaning. It gives more insight into the meaning though:
動詞「占める」の連用形に助動詞「た」が付いた語。「自分のものにした」という意から
占めた is used as "having made one's", as in "having mastered / pwned something or someone". So you could use it when you figured out a problem or when you caught someone in your trap for example.
しめしめ、私の作戦通りだ。 (source: alc)
しめた，この問題なら解けるぞ (source)
As a last note, it can also be used in anticipation for such an outcome:
実際に物事が思い通りに進み始めていなくても、そうなるための方策を思いついたり、人が自分のしかけた罠にかかるといった思い通りに進む兆し・キッカケが感じられたときにもしめしめという
