Rocco Errico seems to be saying that it's useful to look at the Peshitta because Jesus spoke Aramaic and it may give us some insights into idioms and such. There's some truth to this, but a little of it goes a long way. Syriac is a much later, Western Christian dialect and insights into Jesus' language and thought world are much more likely to come from the Hellenistic-Era Qumran Aramaic texts and Jewish Palestinian Aramaic inscriptions. But he seems also to be claiming that the Syriac Peshitta is actually the original Bible, which is simply wrong. It was translated from the Greek and Hebrew some centuries later in a process that is fairly well documented.
The article says that "most theologians agree that the apostles' native tongue was Aramaic, they still say they wrote the New Testament texts in Greek in order to communicate to the emerging Greco-Roman establishment" -- that is, Greek. Actually all philologists (which is what I think they meant, rather than theologians) think the New Testament was written in Greek. There is no controversy about this and anyone who thinks the Peshitta is the original New Testament is, I repeat, simply wrong.
The article spirals off into increasingly odd theological discussions that I'm not going to touch. But I did find this line amusing:
Research shows the Targums and Jewish Palestinian and Babylonian Talmuds were written in Aramaic.First, research on the Targums would show this pretty readily, since anyone who knows Aramaic would notice before long that it's the language they're written in. Second, the Talmuds have a lot of Aramaic in them, but they're largely written in Hebrew.
Lamsa's translation of the Peshitta (which you can access here) could be a useful crib for anyone who wants to read the Syriac Bible but whose Syriac is a little rusty. This might include biblical scholars and other philologists as well as Chaldean and Assyrian Christians. I know next to nothing about him, but the views ascribed to him in this George Lamsa Wikipedia article are crankish, rejected not by most, but by all specialists in Aramaic, and would not be taken seriously in professional journals and monographs.
UPDATE (30 July): Tyler Williams and Christian Brady have already commented on the story. I should have clarified re the first quote above, that it's widely agreed that (apart from Paul) it was the generation after the apostles, or in some cases later people, who wrote the New Testament.
Also reader Gabe Eisenstein writes to say that there's much more Aramaic than Hebrew in the Talmuds (Gemaras). I recall more Hebrew from my very limited work with the Babylonian Talmud, but it's not my area of expertise and I shouldn't have generalized. In any case, "research shows" that there's lots of Aramaic in the Talmuds, but some Hebrew too.