Animate a parametric curve

I would use Manipulate as it is just like Animate but more flexible.

Manipulate[
 Module[{t},
  ParametricPlot3D[{Cos[Sqrt[2] t] (3 + Cos[t]), 
    Sin[Sqrt[2] t] (3 + Cos[t]), Sin[t]},
   {t, 0, maxTime},
   ImageSize -> 300,
   PlotRange -> {{-4, 4}, {-4, 4}, {-1, 1}},
   PlotStyle -> Red]
  ]
 ,
 {{maxTime, 1, "time"}, 1, 50, 0.01, Appearance -> "Labeled"},
 TrackedSymbols :> {maxTime}
 ]

To removed the box and the axis, look at options Boxed -> False and Axes -> False

In Manipulate, you decide which are the control variables in your expression. For each one, you add a control variable definition in the body of the Manipulate. In this case, there is only one control variable.

Animate[ParametricPlot3D[{Cos[Sqrt[2]t], Sin[Sqrt[2]t] (3 + Cos[t]), Sin[t]}, {t, 0, tmax},
   PlotRangePadding -> Scaled[.1], 
   PlotRange -> {{-1.1, 1.1}, {-2 Pi, 2 Pi}, {-1.1, 1.1}}] /. 
    Line -> ({CapForm[None], FaceForm[Opacity[.5, Blue], Yellow], Tube[#, .1]} &), 
 {tmax, 1, 50, .01}, 
 AnimationRunning -> False]