I have never transplanted but have split several if they grow too big & planted those elsewhere. The plant is pretty resilient, just make sure you dig wide enough around to get the small root system.

The plant in the picture you could probably split into 6 decent sized plantings & within a few years will be just as big as the original.

I've divided & transplanted red yuccas for years with 100% success.
They are closely related to yucca, but the genus is different. (Hesperaloe)

Red Yucca grows from clumps of multiple tubers that are easily pulled apart & can be replanted separately.
The plant in your picture could be dug up & broken apart into 5 or more sections and all will grow just fine.
The root system is not expansive, the water is stored in the tuber.

Be sure to plant in full sun for best flowering. The flower stalks will lean towards sun if you plant in the shade. Also beware that while the leaves are deer resistant, the flowers will be grazed heavily by deer.

If you break it up, plant about 3-4 feet apart and keep the soil damp for the first couple of weeks. Water 2x per week for a couple of weeks after that & the plants should be established and never need any additional watering.