if you substitute x = iz
you can split the even and odd terms of the infinite series into cos and sin
e^iz = cos z + i sin z
evaluating at z = pi yields
cos pi + i sin pi
-1 + 0i = -1
the exponential function maps the imaginary axis to the unit circle. pi gets mapped to -1.
This shows its true, but the "why" and real understanding requires calculus and the first week of complex analysis. Otherwise you are just parroting a set of facts.
This presentation does skip some nessasary legwork for complete rigor, but presents a valid non-standard construction of e^ipi.
Specifically, he defines two sets of objects: adders and multipliers, and a function (written e^x for historical reasons) that maps adders into multipliers.
He then generalizes this construction to work in 2 dimensions instead of 1 dimension.
I would add to this construction that e^x is the particular converter that maps pi -> -1. However, I think this requirement is implicit in him stating that e^x is the most "natural" of the converters, and that pi -> -1 is the most natural of mappings.
The only place where you might need calculus is to provide an explicit construction of e^x (and possibly to justify the notational choice of writing it like an exponential).
EDIT:
To make the point clearer. Under the construction presented by the video, e^x is not defined as an infinite some, but rather as a function satisfying certain properties. That this function is equal to a particular infinite sum is a statement that requires proof; and not a statement that is needed for many applications.