It specifically refers to poems written in classical Japanese in non-traditional forms (as opposed to the 5-7-5-7-7 waka and the 5-7-5 haiku) in the Meiji period.
[1] Shintaishi were meant to express emotions, concepts and so on that were seen as too "modern" to be addressed in traditional poetic forms such as haiku and waka.
[1] Yuasa Banketsu's "Jūni no ishizuka" and Ochiai Naobumi's White Aster are usually taken as early representative examples of the form.
[1] The form reached its zenith of popularity at the turn of the twentieth century, specifically the "Meiji 30s", or 1897 to 1906, in Japanese historiographic terms.
[1] In the following decade, however, under the influence of Naturalism and the vernacular free-form poetry movement (口語自由詩運動), it saw a decline and effectively disappeared.