You don’t need a tripod for travel photography, but it can certainly be helpful for specific situations. For example, if you want to take photographs at night or in low light while you’re on holiday.
In addition, a travel tripod can be helpful to make photographic effects like long exposure or adding motion blur. If you find a waterfall when on your travels, you’ll need a tripod to be able to use a slow shutter speed to capture the motion of the water with your camera.
(You could try and find a rock or something stable to rest your camera to achieve the same effect, but you may not find something at the right level to achieve the shot you want.)
Similarly, while travelling, you often come across beautiful night scenes of cities with all the neon lights on – Hong Kong is a good example. To capture a photo like this without having to resort to a high ISO, a travel tripod is useful to allow you to use the right shutter speed and aperture combination.
(Make sure to also check how big of a tripod you need to ensure you’re not craning your head down constantly while taking photos – this can get annoying very quickly, and lead to a sore neck! See also: how heavy should a travel tripod be?)
In conclusion, you can certainly do travel photography without a tripod, but you can take it to the next level if you use one.