The expression “यहां तक कि” means something like “so much so that…”, “so far as…”, “even…”. It is used to give an example of the degree or extent of something. Consider a few examples:
उसको बिल्लियां बहुत पसंद हैं, यहां तक कि उसने एक वेबसाइट बनाई है जो बिल्लियां को समर्पित है – “He really likes cats, so much so that he made a website devoted to cats”
This phrase can be used without कि too:
उसने यहां तक कहा कि मैं उनका सबसे अच्छा विद्यार्थी हूं – “He even said that I am his best student” / “He went so far as to say that I am his best student”