Although this may not answer WHY, a visual I like to use is that of coordinate planes.
Time is a dimension for a reason, it is not a spatial dimension but it still follows the attributes of a dimension.
Lets say you have a simple cartesian plane on your paper. Draw straight across the X-axis at our pretend speed of light, say, 5 mm per second. Going strait across the X axis at 5 mm per second is the fastest possible way to traverse the X coordinate. While doing this, you are not traversing the Y coordinate at all.
If you draw a diagonal line, going up and to the right at 5 mm per second, you traverse both the X and Y coordinates, but it requires you to travel across X more slowly than you were you were not traveling along the Y coordinate. You also in turn travel Y more slowly than you would had you been traveling along Y alone.
Make it three-dimensional with a Z axis and you can still visualize that it is the same. You can traverse all 3 dimensions at once equally at 5mm / second, but this is the slowest possible way to travel among each dimension. You could travel among just X and Y, or Y and Z, or Z and X, without traveling along the missing dimension at all, etc. You are traveling the same speed always, 5mm/second but among each individual dimension the speed changes based on how your speed is "divided" among dimensions.
You can designate a time dimension as well in addition to these spatial dimensions. Look at it this way, if you are traveling ONLY along our time dimension (W) and not moving AT ALL (motionless relative to whatever is measuring your relative time/motion) along our X/Y/Z coordinates then this represents a "normal" flow of time. Start moving in any spatial dimension and it requires that you will be moving at less than 5 mm / second along our W time dimension.
Among our every day speeds relative to what is making measurement we are still moving almost completely along the "time" dimension and almost not at all among the spatial dimension, but rest assured ANY relative motion has an effect on the relative time flow that is measured.
This is also linked to why einstein had the idea that at relative motionless, we are moving through time at the speed of light (from our own inertial frame of reference, we are ALWAYS motionless and that's why we ALWAYS experience time flowing normally according to our own experiences). Move spacially and you are moving slower than the speed of light through time relative to what's measuring you. Move spacially at the speed of light and you cease to move through time relative to what is measuring you. The idea is that our speed through spacetime is ALWAYS at C, but what varies is what dimension we are traveling along.
That's just my way of thinking!
