A short introduction to some ideas of this theory is Kontsevich, M. (1993). Vassiliev's Knot Invariants. Advances in Soviet Mathematics. Vol 16, Part 2. I think that also the following two papers give a general idea of Vassiliev's invariants: Bar-Natan, D. Finite Type Invariants (this is an overview written for the Encyclopedia of Mathematical Physics), Bar-Natan, D. (1995). The fundamental Theorem of Vasilliev's Invariants. Lecture Notes.
If you are interested in a good introduction to these invariants which is written for readers with little background in knot theory, I'd suggest you Chmutov, S., Duzhin, S., & Mostovoy, J. (2012). Introduction to Vassiliev Knot Invariants. Cambridge: Cambridge University Press. Here, some technical results are not entirely proven (they instead refer to other papers for the details); moreover, the book also deals with more advanced topics, so I think that this is a really good reference.