The name ((Trika-Agni-KAla, from which the word TrikAgnikAlaya is derived by chaturthi vibhakti) consists three words: Trika, Agni and KAla.According to the sanskritdictionary.com, Trika means three types. And Agni is of three types:
(Sacrificial) fire of three kinds : Garhpatya. Ahavaniya and Dakshina (गार्हपत्य, आहवनीय and दक्षिण);
The word 'Tika' besides meaning three types has other deep connotations. According to the TantrAloka
trkam sAram chid-IshA -uNmesha-Atmakam, meaning : Trika means Chit(consciousnes), Ichha(Willingness) and Unmesha(Unfolding). These three are also referred to as ParA, aparA, and ParApara shaktis of Lord Shiva according to the Shaiva Philosophy (Paratrishika, pp 602-3).
KAla, meaning time has also three forms known as Tri-KAla : Past, present and future. KAla is also a name of Lord Shiva.
We can conclude that Trika-Agni-kAla means the One Who manifests Himself in three different forms of Fire and Time or as 'Trika', Fire and Time.