Why is the gradient of a curve at a point the same as the gradient of the tangent if you can't use gradient formula on a curve?

This may be initially confusing to get your head around but gradient is really just an expression of "rate of change" on a curve the rate of change is always changing at every point, this about it like climbing a mountain there are going to be steeper parts and and flatter parts so the slope or " rate of change" is going to be different at every point. While on a straight line like a ramp the gradient or 'rate of change' is constant for the whole line.
So how do the two relate? well when want to know the gradient of a point on a curve, we can think of zooming in so close to that point that it looks like a straight line and you take two points that are really really close together in the x axis, and then look at their y values and use the gradient formula for a straight line. because these points are so close together it's the same as saying they can be evaluated as one point. and if you imagine these two close points had a straight line drawn through them and extended along then the gradient of that line will be the same for any two points on that line and therefore will be the same as the gradient of the curve at that point we want to evaluate it at.