Most case, "다니" is conjunction and has similar use as "~다니까". it implies the meaning of "because" or "since". In this context, "It is such a pleasure that(because) I am able to speak Korean to you."; "다니" is equivalent to "that" or "because". e.g.) "그가 벌써 떠난다니 이상하다." will be "it is weird that/ because he is already leaving." Hope it makes sense to you.. let me know if you have further questions. :)

"-다니"는 뜻밖의 놀라움을 나타내는 말입니다. It's a great pleasure that I can talk with you in Korean.