Often the hyperbolic trigonometric functions are just treated algebraically, as compositions of exponential functions. However there is also geometry to them. Just as sine and cosine are defined for the unit circle, sinh and cosh can be defined for the unit parabola.
The most difficult thing to understand is what the input of these functions is. It is described as the "hyperbolic angle" but it really is the area of the hyperbolic sector, symmetric across the x-axis, with endpoint a point on the parabola.
This also has a connection to circle trigonometry, as we could think about the area of the sector determined by a point on the circle.