Does anyone know why my arrow turns into this strange shape in Inkscape when I fill it?
Here is my svg file from Inkscape.
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The object has a fill, and that is the problem.
What you are seeing is what happens when you try to fill an open path. This is not just specific to Inkscape, the same happens in other vector software, such as Illustrator. There's nothing strange going on here at all, it's normal behaviour. However, you shouldn't apply fills to open paths if you want to avoid this kind of thing.
To fix it, just select the path, and remove the fill.
Same thing happens in Adobe Illustrator, same fix
If your goal is to change the colour of an arrow, then redraw the line, add an arrow head marker, and change the colour of the stroke (not the fill).
An arrow is just a curve with an arrow head at the end. Other than that, it behaves like any shape. When a shape is filled, the area between its curves is flooded with ink.
Your shape (and others, like the spiral) is special in the way that the curve is not closed. In that case, Inkscape will connect the beginning point of the curve and the end point of the curve with a theoretical (thus invisible) line.
The following comparison might make it clear: you see the curve in black, the theoretical line dotted and the fill in blue.