The mechanism for electromagnetic (EM) mode transition and filtering in an asymptotically single-HE11-mode hollow THz Bragg fibre is investigated. We designed, fabricated, and measured Bragg fibres with an asymptotically single-mode pattern, achieving measured signal propagation loss of better than 3 dB/m at 0.265 THz and with an operating frequency range from 0.246 to 0.276 THz. Mode transition and filtering effects are both verified by 3D full-wave simulations using measured material properties, with geometrical parameters extracted from the fabricated Bragg fibre prototypes. By optimizing the coupling efficiency between the free-space Gaussian beam and the guided Bessel function mode, the optimum distance of mode transition from the Gaussian-beam excitation into the guided mode is calculated to be ~13.7 free-space wavelengths in our fibre, to ensure fast EM-field convergence to the desired asymptotically single-mode mode pattern in the Bragg fibre. After this mode transition region, the electric field amplitude ratio between the desired HE11 mode and the main competing HE12 mode is approximately 7 times, with the HE12 mode attenuation being more than 10 dB/m larger than the fundamental HE11 mode; the results indicate that our fibre is one of the best candidates for low-loss asymptotically single-mode terahertz signal interconnections.