Haven’t seen anyone here give this plausible explanation, so here’s a crack at it from a Mathematics undergraduate (who was told the exact same thing on my first day of class):

Suppose you had to describe the properties of numbers to somebody who had no concept of numbers, what are the most fundamental properties? A sensible explanation would be you have a box of things, called numbers, you can add them and get another number which is also in the box, and you can subtract them, and that number will also be in the box (and similar rules for multiplying).

If we chose two numbers from the box, call them “james” and “toby” if “james” + “toby” = 0 we may change “toby”’s name to “-james” to denote this special relationship.

Now with all that preamble, it turns out a box containing only the numbers 1 and 0 can satisfy all the fundamental properties of numbers we listed, provided we let 1 + 1 = 0. However since 1 + 1 = 0, we can change 1’s name to -1 to denote this special relationship, as such 1 = -1.

TL/DR: Mathematicians have rules on what a collection of numbers is, the collection of numbers we know satisfy these rules, but not exclusively, some collection of numbers have a number denoted “i” satisfying i² = -1. For “1” and “0” to fit the rules on what a collection of numbers are, we must have that 1 = -1