Here's a visualization of a hyperbola with a horizontal transverse axis centered around the origin. Its formula is x2/a2 − y2/b2 = 1. Using the sliders you can alter a and b, the distances from the origin to each vertex or covertex respectively. (Or, put another way, 2a is the length of the transverse axis and 2b is the length of the conjugate axis.) c is defined in the usual Pythagorean manner: a2+b2 = c2.
The diagonals of the 2a × 2b rectangle centered on the origin form the asymptotes. Do you see why the three segments labeled c are the same length? Do you see why a2+b2 = c2? The circle centered on the origin may help.
The defining characteristic of a hyperbola is that the distances between a point P on the hyperbola and the two foci differ by a constant. In this case, the constant difference is 2a. Look at the circle centered on point P and the circle centered on the focus F1. Circle F1 always has radius 2a regardless of the position of P, representing the constant difference. Circle P has a radius that varies as P moves, but its radius is always the distance between P and F2. Thus you can see the constant difference between the distances PF1 and PF2.
Try it! Press the "Open Geogebra" button; then adjust the a and b sliders and see what happens. Also move point P along the hyperbola.