Why is Clojure dynamically typed?

I agree, a purely functional language can still have an interactive read-eval-print-loop, and would have an easier time with type inference. I assume Clojure wanted to attract lisp programmers by being "lisp for the jvm", and chose to be dynamic like other lisps. Another factor is that type systems need to be designed as the very first step of the language, and it's faster for language implementors to just skip that step.

If a program is compiled you can't change it anymore.

This is wrong. In image-based systems, like Lisp (Clojure can be seen as a Lisp dialect) and Smalltalk, you can change the compiled environment. Development in such a language typically means working on a running system, adding and changing function definitions, macro definitions, parameters etc. (adding means compiling and loading into the image).

This has a lot of benefits. For one, all the tools can interact directly with the program and do not need to guess at the system's behaviour. You also do not have any long compilation pauses, because each compiled unit is very small (it is very rare to recompile everything). The NASA JPL once corrected a running Lisp system on a probe hundreds of thousands of kilometres away in space.

For such a system, it is very natural to have type information available at runtime (that is what dynamic typing means). Of course, nothing hinders you from also doing type inference and type checks at compilation time. These concepts are orthogonal. Modern Lisp implementations typically can do both.

Well first of all Clojure is a Lisp and Lisps traditionally have always been dynamically typed.

Second as the excerpt you quoted said Clojure is a dynamic language. This means, among other things, that you can define new functions at runtime, evaluate arbitrary code at runtime and so on. All of these things are hard or impossible to do in statically typed languages (without plastering casts all over the place).

Another reason is that macros might complicate debugging type errors immensely. I imagine that generating meaningful error messages for type errors produced by macro-generated code would be quite a task for the compiler.