A line is a 1 dimensional object because it can be formed from scalar multiples of a single point (sometimes called a vector) in 2-space, 3-space and so on.

Imagine what would happen when you take the coordinates (ta,tb,tc) in 3-space and let t vary through the real number line while a,b,c are constant real numbers. It’s not hard to imagine that if we were to plot this set of points for every t in the real numbers we’d get a straight line through the origin. Yet, we need to vary only a single value. In this sense the number of values we must vary, also called parameters, to write our object in coordinate form like this gives you the dimension.