Tea drunkenness is a somewhat controversial topic. Some people compare it to a strong caffeine buzz, and say their side effects include shakiness, nausea, strong dizziness and other unpleasant sensations associated with consuming excess caffeine. However, most serious tea aficionados describe their experiences with being 'tea drunk' very differently.
For me, being tea drunk feels euphoric. There is a strong sense of focus and calm, an acceptance of the world around me / feeling of contentment, and a pleasing lightheadedness / sense of floating. I've heard others list symptoms such as a giggly, bouncy or goofy feeling, a feeling of emotional bliss, a contemplative or philosophical mindset, or an introspective, sensitive mood. The overall effect is often describes as a very 'Zen' feeling.
It is thought that tea drunkenness is a result of the combination of caffeine and other molecules in tea, particularly L-theanine. Consumed without caffeine, L-theanine induces a very calm (and often very sleepy) state, and it is in trial stages for use as an anxiety reducing drug. However, when combined with caffeine, L-theanine is thought to increase the production of alpha waves in the brain, inducing a meditative state without making you tired.
Observing the physical, mental and emotional effects of a tea is just one of the many aspects of tea tasting and appreciation. To attain a state of tea drunkenness, start by eating some food to cushion your stomach to the tea's caffeine and slow the absorption of tea's natural chemical compounds. Then, prepare your tea (preferably a high quality oolong, shade-grown Japanese green tea or aged pu-erh*), savor it, and note how you feel.
* Some say that you can only get tea drunk by drinking high quality tea. (In my experience, this is the case.) Some people find that they get tea drunk more easily from shade-grown Japanese green teas (like Matcha Green Tea and Gyokuro Green Tea), aged pu-erh teas and high quality oolong teas (especially Tieguanyin Oolong).