The claim of “mass increase with velocity” is based on using the word “mass” for something that is not a property of the object itself but rather depends also on the relative velocity of the observer.
Although this usage is not “wrong” and was fairly common in the early days, most physicists soon chose to use the word “mass” just for the invariant “rest mass” and followed the title of Einstein’s initial paper on the effect by just using the word “inertia” for the velocity-dependent quantity that used to be called the “inertial (or relativistic) mass”.
In order to prove that the observed “inertia” depends on the relative speed (and direction) of motion of the observer it is necessary first to define it.
One way is to define it as the ratio of applied force to observed acceleration, but this in turn requires a clear definition of force. For example you could perhaps look at the dynamics of a collision with an object of known mass at rest in the observer’s frame.
Or perhaps you could just read Einstein’s original paper on the subject.