Parser contains the main lexing & parsing code. The main functionality is taking a piece of text and parsing it into an abstract syntax tree (AST). You can parse a whole .m file as well as one expression.
Parser.Tests contains some tests for the parsing code. For example, ThereAndBackAgain.TestEverything() parses all the files in MATLAB's toolbox folder (you'll need a MATLAB installation to do this) and checks that the text constructed from the resulting ASTs is the same as input text.
Semantics contains some attempts at the next step: inferring semantics of the code from ASTs. Examples may include tracking variable usage, external function calls, etc.
SyntaxGenerator is a stand-alone console program that generates a part of parsing code (mostly class definitions for syntax nodes) from a simple XML. We use it to avoid writing tons of boilerplate code by hand.
ConsoleDemo is a console project used to play with parsing, print the resulting ASTs (possibly with syntax highlighting), etc.

The parser architecture is inspired by Microsoft's Roslyn compiler for C#. Studying Roslyn source code is a great way to learn about lexers, parsers, and all other parts of a compiler.