the technical math definition is a rigourous formulation that encapsulates exactly the same thing as what we mean when we say things are curved, but one that also extends far more generally into contexts where our old intuition fails.
The same is true for most mathematics. For example, we are introduced to multiplication as repeated addition: 3x == x + x + x or 2x == x + x and more generally nx == x + x + ... + x, for n number of times. Of course this is only defined over naturals, what would it mean if we instead took n to be fractional, negative, irrational, or even complex? We of can easily generalise multiplication over larger and more complex fields and spaces, but in doing so we must abandon our old intuitive idea that nx is x + x n-times.
The same is true for most mathematics. For example, we are introduced to multiplication as repeated addition: 3x == x + x + x or 2x == x + x and more generally nx == x + x + ... + x, for n number of times. Of course this is only defined over naturals, what would it mean if we instead took n to be fractional, negative, irrational, or even complex? We of can easily generalise multiplication over larger and more complex fields and spaces, but in doing so we must abandon our old intuitive idea that nx is x + x n-times.