One it is an altered state of mind, which can be taken several different ways. When in a peaceful or familiar setting, being teadrunk will raise your awareness while calming you at the same time. Its really a great feeling and time seems to slow down. I also feel emotions get enhanced, so if you are happy you will be happier than you really have any reason to be (besides of course, the obvious fact that you just drank good tea).
Though the enhanced emotions can have a down side, as I tend to be a little anxious in public when I am alone or not with people I know, and should I venture out somewhere after having a lot of good tea, it can be a mess, in the sense that I know what I need to get done, and I want to get it done quickly. But when nervous and anxious it can lead to me almost being counter productive.
Two, it happens through drinking, and as the name implies drinking tea, so literally you are drunk on tea.
Three, tea can have different strengths, the term used to describe a teas strength at getting yourself teadrunk is in my mind the equivalent to often used, though hard to understand term Cha qi, or rather the life energy of the tea.
Four, Somedays it can take very little to put you over the edge, while other days it can take quite a bit.
So how does one get teadrunk? To everyone that brews western style, I am not sure it will be possible for you to experience, at least not without drinking 6 or 8 cups of very good tea. But the sure fire way is to brew good quality tea, and brew it strong. I sometimes note that a single session of Koicha can be enough to put me over the edge, and it often is. A general rule of thumb is aged teas tend to put forth more qi, than new teas, and High mountain or Old tree more than standard plantation, but really you drink enough of any tea and in my mind you should be able to get to some level of teadrunken-ness .
So I'll leave you with these parting words.
If you must get drunk, get teadrunk.