Answer:
While it has been traditional in most Martial Art schools to train bare-footed, many schools are adopting new traditions. Typically, Judo schools that train on thicker mats still practice without shoes, and a lot of modern Taekwondo, Karate, and Aikido schools still do not permit shoes on the training floor.
However, some schools will permit specialized training shoes that are carried in clean, and worn only for training purposes. The Asian custom of removing your shoes when entering any building was for both health reasons, and cultural philosophy. There is often times dirt, and sharp objects tracked indoors on the bottom of shoes. In third-world countries where serious illnesses often lead to death, people do not share public toilet seats, and they remove their shoes before entering a home, a restaurant, or other businesses.
Shoes worn outside are also believed to be contaminated spiritually, and are removed before entering temples, and other places of worship or spiritual training. In Asian culture, the shoes are said to carry your troubles indoors with you, so they should be left outside, or at the door. However, many buildings in Asian countries are not heated very well, if at all, so indoor slippers, or thongs are often provided for guests. Martial Art shoes that are not worn outside help to keep the feet warm, prevent the spread of germs through contamination of the feet, and can facilitate pivoting and kicking on a bag or hand-held target.
Most people wear shoes in their daily lives, so wearing shoes in training is becoming more popular and acceptable, however beginners are still encouraged to spend a great deal of time bare-foot in order to learn the proper foot positions in those arts that use kicks, and it tends to build calluses to toughen the feet. Keep in mind, that modern Martial Art classes are not exclusive to the dedicated, die-hard warrior. Many men, women, and children want to learn how to defend themselves, but are not interested in having callused body parts for the rest of their lives.
Some students wear martial arts shoes to provide the foot a bit of additional support, especially those suffering from plantar fasciitis or various low back problems. A little support can significantly increase their comfort, particularly during warm-up activities such as running.
Also, many modern-day tournaments require the judging officials, and referees to wear specialized shoes as part of an official uniform (usually a specific color of shirt, tie, pants, and shoes). Another purpose for a comfortable shoe that adapts well to the Martial Art setting is that many instructors spend the entire day at their school, which is essentially their workplace. Besides teaching classes, there are cleaning chores, office work, and interviews with prospective clients. During those times, it is more comfortable, and appropriate to wear shoes, then remove them as the instructor steps onto the training floor or mat for teaching.