Why does this regex_split:nnN results in error and insertion of `{`?

You are adding a spurious set of braces in your prop_put:Nnn command:

prop_put:Nnn l_files_prop {#1} {{#2}}

Changing that to

prop_put:Nnn l_files_prop {#1} {#2}

solves the problem. The error happens because with a token list {hello;world}, splitting it at the ; will leave unbalanced braces in each item ({hello and world}), which is illegal, so l3regex inserts some to balance the token lists.

I also changed some things to conform better with expl3 standards:

• All variables should be declared, so you need a tl_new:N l__files_split_tl and a tl_new:N l__files (see next item about this one);
• Variables should be named ⟨scope⟩_⟨module⟩_⟨name⟩_⟨type⟩ so l__files is a terrible name. Something like l__fabian_files_tl is better, assuming fabian is the module;
• cs_generate_variant:Nn can be done only once per variant, so you can move it outside the command;
• Assuming #2 is a control sequence, use:c {cs_to_str:N #2} has the same effect as just #2, so you can simplify;
• The output of regex_split:nnN is a seq variable, not a tl, so you should use that instead;
• [bonus] If you just want to split at a ; you don't need regex at all. Instead of the regex_split:nVN line you can use seq_set_split:NnV l__fabian_files_split_seq { ; } l__fabian_files_tl (the main difference is that regex_split:nVN will preserve spaces around items, but seq_set_split:NnV will trim spaces).

Here's the code:

documentclass{article}
usepackage[T1]{fontenc}
usepackage{xparse}
usepackage{amsmath, amssymb}

begin{document}

ExplSyntaxOn
prop_new:N l_fabian_files_prop
tl_new:N l__fabian_files_tl
seq_new:N l__fabian_files_split_seq
% cs_generate_variant:Nn regex_split:nnN { nV }

NewDocumentCommand addConfig { mm }
  { prop_put:Nnn l_fabian_files_prop {#1} {#2} }
NewDocumentCommand printConfig { mm }
  {
    prop_get:NnN l_fabian_files_prop {#1} l__fabian_files_tl
    % regex_split:nVN { ; } l__fabian_files_tl l__fabian_files_split_seq
    seq_set_split:NnV l__fabian_files_split_seq { ; } l__fabian_files_tl
    seq_map_inline:Nn l__fabian_files_split_seq { #2 {##1}  }
  }

ExplSyntaxOff

addConfig{config1}{hello; world}
addConfig{config2}{bye; world}
addConfig{config3}{testFile.tex; testFile2.tex}

noindent
printConfig{config1}{textit}
printConfig{config2}{textbf}
%printConfig{config3}{input}

end{document}