You can do this outside of the manipulate box and show the variable's value as a legend. Also by using fixed image sizes and aspect ratios you can minimize the occurrence of black bars as the variables change.
First Step:
Make a table of your desired Plots, I think the code here is relatively easy to understand:
Anim = Table[
Plot[Sin[x], {x, 0, t}, PlotRangePadding -> 0.1, ImageSize -> 500,
AspectRatio -> 1/2, PlotLegends -> TextString@Row@{"T=", t}]
, {t, Pi, 2 Pi, .1}
];
Second Step: Export it as a GIF image :)
Export["anim.gif", Anim]
Final Result:

Hope it's suitable for what you have in mind ;)
UPDATE
If you want it to move forward an then backward you can make tables of the animation for each direction and join them:
Anim1 = Table[
Plot[Sin[x], {x, 0, t}, PlotRangePadding -> 0.1, ImageSize -> 500,
AspectRatio -> 1/2, PlotLegends -> TextString@Row@{"T=", t}]
, {t, Pi, 2 Pi, .1}
];
Anim2 = Table[
Plot[Sin[x], {x, 0, t}, PlotRangePadding -> 0.1, ImageSize -> 500,
AspectRatio -> 1/2, PlotLegends -> TextString@Row@{"T=", t}]
, {t, 2 Pi, Pi, -.1}
];
Anim = Join[Anim1, Anim2];

UPDATE 2
If you want to do this with the manipulation box just add a two-dimensional picture size to your plot like this:
man = Manipulate[
Plot[Sin[x], {x, 0, end},
ImageSize -> {500, 250}], {end, Pi, 2 Pi, 0.1,
Appearance -> "Open"}, ContentSize -> 550]

Or this one (without fixing the plot range):
man = Manipulate[
Plot[Sin[x], {x, 0, end}, ImageSize -> {500, 250}], {end, Pi, 2 Pi,
0.1, Appearance -> "Open"}, ContentSize -> 550]

+
toggler to open the additional control bar? It includesInputField
which shows how the value updates. $\endgroup$ – e.doroskevic Aug 11 '16 at 12:55{end, Pi, 2 Pi, 0.1, Appearance -> "Open"}
. $\endgroup$ – Kuba♦ Aug 11 '16 at 12:56ImagePadding -> 10
, now I get this. Another option would be to giveImageSize
two numbers, so that the width and height are always specified. $\endgroup$ – Jason B. Aug 11 '16 at 14:16