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First, picture a two coordinate grid. (The kind you used to plot equations in school). A length of exactly one whatever, call it a foot, lying horizontal takes up one foot horizontal and no space vertical. The same length vertical takes up on vertical foot, but no horizontal space.
Take that length, and lay it at an angle. Now, it takes up space both vertically and horizontally. You are familiar with perception in two dimensions, and so you can easily see this. If the length tilts more towards the horizontal, then it takes up less vertical space and more horizontal space. However, at all times the length is exactly the same.
Now, imagine that acceleration can be plotted onto a coordinate grid instead of a vertical spatial measure. After all, time is a measurable dimension, and acceleration and speed are simply functions of time compared to a physical distance. So call this new coordinate (y*t). You are unfamiliar with perception into this new dimension, and so your brain attempts to explain them in terms of your familiar three coordinate system. However, when it does this, things don't sync up perfectly.
A length at rest (that is, horizontal) appears exactly the same in this new grid as it did in the old grid. When you start applying acceleration to that length (tilting it up), It starts to take up less space on the x coordinate and more on the (y*t) coordinate. Your brain will continue to think of this coordinate in spatial terms, and so will be unable to intuit why is doesn't act the same as plain old y.
This mental grid helps to explain temporal dilation as well. An object exactly at rest (perfectly horizontal) would be experiencing no time. This is effectively impossible (or performed only by things like photons; take your pick). Objects subject to a greater acceleration experience more personal time than those subjected to less acceleration, or, alternatively, experience less distance.
This isn't an exactly correct way of describing relativity, but perhaps it's a helpful mental exercise nonetheless? Hopefully at some point your brain will just click, and then you'll understand it fine.
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