Mathematical background for TQFT I am physicist. I`ve started studying Topological QFT.
What would you recommend to read in mathematical field for understanding Witten’s old articles of 80s-90s? 
What books/articles could help form proper mathematical background for TQFT?
 A: So we are talking Quantum Field Theory and the Jones Polynomial and Chern–Simons Gauge Theory as a String Theory. My advice is to start reading Witten and pick up background on what he uses as you go along, trying to learn all those mathematical theories in their entirety may be insurmountable.
Bar-Natan's thesis is a detailed interpretation of Witten's perturbative Chern–Simons theory by a mathematician, see also other papers on his website. It brought the theory of finite type or Vassiliev link invariants into prominence. For a more recent take look at Sawon's paper.
Freed's survey is a nice segway from the Chern-Simons theory to modern mathematical TQFT. For Witten's non-perturbative approach the main mathematical ingredients are knots and links, skein relations, polynomial invariants, surgery on links that produces 3-manifolds, connections, holonomy and conformal blocks. The best introduction to knots is Adams' Knot Book, it does cover Jones polynomial and skein relations, but to get into surgery you'll need Rolfsen. I am hoping that you know connections and holonomy (gauge theory) from the physical side, I liked this book. Representation theory of semisimple Lie algebras/compact Lie groups, $SU(N)$ in particular (roots, weights, Weyl character formula, etc., see here for example or in this standard text), should be more or less enough to understand Witten's use of conformal blocks. Conformal blocks themselves can be interpreted either in terms of bundles on the moduli of curves or of integrable representations of affine Lie algebras, but either path is very convoluted, perhaps references here will help.
However, reading Witten is different from learning (modern) mathematical foundations of Chern-Simons theory. Non-perturbative approach is based on quantum groups and graphic calculus of Reshetikhin-Turaev (which is a good thing to know since it's pretty universal to 2D TQFT), the standard reference is Turaev's book, which also goes into TQFT in general, see also part II here. Understanding quantum groups and computations with them also requires knowing representations of Lie groups.
