It’s because you animated the spline and not the Extrude containing it but the texture is attached to the Extrude, so the spline moves through the texture space (which it’s getting from the Extrude) and the texture slides.
Try clicking on any one of the texture tags and use the Texture Tool or the Texture Axis tool. You will see the relevant letter move into the texture ‘cube’ over the first 85 frames or so, now try putting the texture on the spline and now the texture space should move with the letter.
The problem is that won’t texture the letter correctly so I think you have to redo by animating the Extrude and not the spline in it. Making it editable would solve the texture space thing as well but that breaks the way you animated so that won’t work.