For the equation of motion for motion relative to a rotating coordinate system the coriolis force in the equation of motion is at right angles to the current velocity.
The coriolis force in the equation of motion represents acceleration with respect to the rotating coordinate system. Since the coriolis force is at right angles to the current velocity the acceleration due to the coriolis force is at right angles to the current velocity.
The two velocity components $u$ and $v$ are defined as perpendicular to each other.
First the two simplest cases:
velocity in the $u$ direction. The acceleration due to the coriolis force is at right angles to that, hence acceleration in the $v$ direction.
velocity in the $v$ direction. The acceleration due to the coriolis force is at right angles to that, hence acceleration in the $u$ direction.
But of course in general the current velocity will be at some angle in between the $u$ direction and the $v$ direction. So: you decompose the current velocity in a $u$ component and a $v$ component. That gives you two acceleration components, and when you recombine those acceleration components you get an acceleration that is at right angles to the current velocity.
It may be that you are actually wondering why the coriolis force is at right angles to the current velocity. The reason for that is mathematical.
When you use a rotating coordinate system it is convenient to decompose the acceleration relative to the rotating coordinate system in the following two components:
an acceleration that is a function of your distance to the axis of rotation (referred to as 'centrifugal force')
an acceleation that depends on your velocity relative to the rotating coordinate system (referred to as 'coriolis force').
In the case of constant angular velocity of the rotating coordinate system those two combined are exhaustive, so you have everything covered.
When you do the above decomposition in 'centrifugal' and 'coriolis' it follows mathematically that the coriolis force comes out at right angles to the current velocity. That is not a physics thing, it's a mathematical thing.