The anoixantaria are indeed a musical setting of the last few verses of Psalm 103/104. They are (to my knowledge) chanted only in Plagal of the Fourth -- and only when a Hierarch is present.
A "terirem" -- I think that is a mis-spelling -- is said to be the sound the angels make. There is a whole elaborate Byzantine theology (of course!) of the terirem, but, musically speaking, it's a space filler: A chanter uses it in the long, fancy settings of hymns (like the Cherubic hymn) to play with the scale, show off and take up more time. In monastic settings, it just takes up time (gotta fill up those 9 hours for Vigil somehow!). I think when I was on Mt. Athos we sang various rems for about 35 mins straight.
Technically, the Anoixantaria should only be chanted for the feastday of the Church (whichever day that is), even though they're also chanted for vigils and sometimes for secondary feasts that are very important to a parish. They are the concluding verses to psalm 103 (beginning with "When you open your hand they are filled with good things." Anoixantos = when you open, which is where the name comes from) which, depending on the musical composer, will also have praises of the Holy Trinity added to the verses.
As for the Kratema (which literally means "hold") - this is the technical name for the te-ri-rem, ne-na-no, and other "filler" phrases. One interpretation is the angelic one that pensateomnia mentions. Another that the monks have said is that it is the lullaby that the Panagia used to put Christ to sleep. As a corollary to the Angelic explanation - you're right, the simplicity is supposed to be humbling - if Christ is able to be praised with words that make no sense, but are instead simple syllables repeated over and over, then our over-complicated hymns and genuflections and whatnot are a bit much, no?