I need to know what is exactly DHE_EXPORT cipher suite and why it is still supported by some servers. Is there any implementation or details about this weak crypto?
Export ciphers date back to the time when cryptography export was regulated by governments, most notably the US government (see this answer for a good overview of the history of export restrictions). Back then, exporting cryptography with more than 56 bit strength was forbidden. For the Diffie-Hellman Key Agreement scheme, 56 bit of security meant that they could only use 512 bit parameters (I'm going to assume that you know how Diffie-Hellman works. If not, see for example here).
All of the remaining details of the protocol are, as far as I am aware, identical to regular Diffie-Hellman. They are a relic from past times that continually crop up and cause trouble (like in the case of logjam, which is explained well in this answer and this article). There is absolutely no good reason to support them in this day and age, and if you find a server that still allows it, you should contact their admins and tell them to disable it.
(This is also a very good reason why we should not go back to intentionally weakening cryptography, as some people have proposed lately - we tried that once and it's still hurting us, over 20 years later.)