How can I display Latex style font using Jupyter Notebook Extension in VS Code?

That is the correct LaTeX output for that expression. That is the result of the integral displayed in terms of the hypergeometric function:

https://en.wikipedia.org/wiki/Generalized_hypergeometric_function

You might not recognise the function but it is an ordinary mathematical function and you can substitute values into is and evaluate it etc:

In [17]: antiderivative = integrate(Si(x)/x, x)

In [18]: antiderivative
Out[18]: 
       ⎛              │   2 ⎞
   ┌─  ⎜   1/2, 1/2   │ -x  ⎟
x⋅ ├─  ⎜              │ ────⎟
  2╵ 3 ⎝3/2, 3/2, 3/2 │  4  ⎠

In [19]: antiderivative.subs(x, 1)
Out[19]: 
 ┌─  ⎛   1/2, 1/2   │     ⎞
 ├─  ⎜              │ -1/4⎟
2╵ 3 ⎝3/2, 3/2, 3/2 │     ⎠

In [20]: antiderivative.subs(x, 1).n()
Out[20]: 0.981810799391358

Many ordinary looking mathematical functions can be expressed in terms of hypergeometric functions and sometimes it is possible to simplify them into something more recognisable:

In [27]: hyper([], [S(1)/2], -x**2/4)
Out[27]: 
     ⎛    │   2 ⎞
 ┌─  ⎜    │ -x  ⎟
 ├─  ⎜    │ ────⎟
0╵ 1 ⎝1/2 │  4  ⎠

In [28]: hyperexpand(_)
Out[28]: cos(x)

It is useful to be able to rewrite an integral in terms of hypergeometric functions because a routine that can integrate hypergeometric functions can work for a wide variety of possible integrands. It is particularly use for special functions (such as Si) without needing special rules for each new function that we might want to integrate. SymPy has a specific integration routine meijerg which does this using the even more general Meijer G function:

https://en.wikipedia.org/wiki/Meijer_G-function

SymPy has used the meijerg routine for this integral although it looks like the result has been converted to hypergeometric functions rather than G functions. Sometimes it is possible to simplify the result of a definite integral even if it is computed using an antiderivative that can only be expressed in terms of hypergeometric/G functions.

In the case of this integral though it doesn't look like SymPy can express it using other functions. I checked WolframAlpha as well which gives a less simple (but equivalent) representation in terms of hypergeometric functions as well:

https://www.wolframalpha.com/input/?i=integrate+Si%28x%29%2Fx

I don't know why you've got two different types of output (Unicode and LaTeX), but I can tell you how you can be sure that the LaTeX rendering is the unfailing default

from sympy import *
init_printing(use_latex=True)
...