The video here by floatheadphysics says that Mercury precession is caused by curvature in spacetime and can be represented by space turning into a cone and going in another dimension (which don't seem to be time)
So can this effect be visualised without involving 5th dimension?
space turning into a cone and going in another dimension
In the example the cone is still a 2D manifold. The geometry that is being described is the 2D geometry of the cone itself, not the 3D geometry in which the cone is “embedded”.
So can this effect be visualised without involving 5th dimension?
Gravity in general relativity is mathematically described using 4 dimensions (3 of space and 1 of time). In the math there is no need for any embedding space. Such a space, if it exists, has no bearing on the physics in the 4D manifold.
As far as visualization goes, we have trouble visualizing even 4 dimensions. I don’t think adding a 5th would help with visualization.
The cone idea is offering a way to get intuition about the effects of curvature on a two-dimensional space. In this example the two-dimensional space is a slice through three-dimensional space at some given moment in time (where we adopt the time coordinate in a sensible way). You can think of it as a plane containing the center of the Sun and all of Mercury's orbit. Then the Newtonian physics will predict some precession (owing to influence of Jupiter etc.) and General Relativity (GR) will predict a further contribution to the precession. A good insight into this GR contribution is to note that the circumference of a circle is not equal to $\pi$ times its diameter when the space is curved. In this example the circumference is a little less then $2\pi r$ so after one orbit Mercury arrives back where it started a little early compared to Newtonian expectations.
The cone model is giving you a way of visualizing how curvature can lead to this kind of result. If you regard it as a visual aid rather than as a statement about a 5th dimension then you will not worry about 5th dimension. You will treat it as an aid to intuition. It is an aid which can be made mathematically precise.
