The sense of the English word "integral" that relates to the mathematical usage is the adjectival meaning "whole". From Wiktionary:
Constituting a whole together with other parts or factors; not
omittable or removable
Something not omittable or removable can easily be seen to be necessary or essential.
The mathematical notions of "integers" and "integrals" relate to this meaning. Another term for integers is "whole numbers", in the sense that they have no fractional part. They don't have pieces--they are whole.
Integration in math is the process of summing up (in the limit) small pieces or slices of a shape (represented as the area under a function curve) into a whole area. In other words, from the pieces a whole is made.
The mathematical usage derives from the original meaning of "whole".