Subtle means that something exists, but that it is not obvious. It has a sense of being delicate, not immediately noticeable, indirect.
For example, imagine two people walking into a room and greeting their friends. The first person walks into a room and loudly shouts "Hello! I am here!" The second person briefly smiles at everyone and says nothing. The second person is being subtle.
It is the same basic idea with the specific example you gave. Irony being "subtle" here means that a person might not immediately notice the joke upon hearing it. It is a less obvious way of saying something funny.