He explained that this is a level of prayer achieved only by those people who have a nefesh from the world of Atzilut. When one is on such a level, there is no difference between one's thought, speech and action, they are all united. At such a level, one can pray like a one-day-old child.
Before I got a chance to speak to my Rav, I explored a little in my own tefilla, trying to imagine and understand what it is to pray like a new born child. Since my son is only eight weeks old, I looked to him for examples. When a baby cries, really cries, they become so worked up and overwhelmed that their whole body becomes tense. They become so caught up that the reason they started crying doesn't appear to matter, they just cry until they're exhausted or until someone intervenes.
It's such an exhausting avodah to try and pray in some semblance of the immediacy of a baby's needs. My tefilla went from around a half hour shemonah esrei to a three minute urgent shemonah esrei.
Somewhere in the middle of my explorations was a form of tefilla that is closer to "children's tefillah" than to a one-day old baby's tefillah. It was a sort of joy arrived at simply because being involved in tefillah was fun. It felt somewhat like being one of those sunshine-y kids who just spread happiness with them wherever they are.
Anyway my Rav said that the Rebbe m'Komarna brings down that there's another form of prayer, the simple "prayer of women and children" which is just that, simple, unassuming, and genuine. That works wonders in heaven and completely confounds the ministering angels. This is something that anyone can achieve--whereas the prayer of a day-old child is a spiritual level that is virtually beyond comprehension.
One last point, the splitting of the sea in the exodus from Egypt is said to be akin to the birth of a child. So, in that sense, Shirat HaYam (אז ישיר) is the cry of the just-born child. Chazal says about Shirat HaYam that it is white fire on black fire, whereas the rest of the Torah is black fire on white fire. So these I take to be clues about the tefillah of a newborn child.