The “underwear” you refer to is a temple garment, often called simply “garments” by Latter-day Saints. Only adult Mormons who have made temple covenants wear them. They are an “outward expression of an inward commitment to always remember the Lord.”
Mormon temple garments are similar to normal underwear. Tops for women come in various fabrics and necklines, all of which are as low as the bra line. Women’s tops have cap sleeves, except for a special thermal garment for use in cold weather. Bottoms either end just above the knee or mid-calf to wear with pants. Men’s garments are also usually two-piece. The top looks very similar to a normal tee-shirt, and the bottom has a similar look to jockey shorts, but ends just above the knees. Garments are white, but there are special garments for members of the armed forces, which may be khaki or black. Small symbols are woven into the garment, which represent temple covenants. They remind the wearer of his commitment to Christ, just as wearing the cross does for Christians of other faiths. Rumors of the magical qualities of temple garments are just that ─ rumors. Garments are meant to be protective, but that protection is understood to be spiritual, rather than physical. There have been experiences of physical protection, but those should be attributed to the spirituality of the wearer.
Mormons who have made temple covenants wear their garments day and night. They remove them for demanding exercise, swimming, dance performance, personal care, and intimacy. The garment is not to be altered to accommodate fashion. American temple-attending Mormons at a summer carnival might be seen wearing Bermuda shorts and short sleeved tee shirts, but not short-shorts and sleeveless tops. “The wearing of such a garment does not prevent members from dressing in the fashionable clothing generally worn in nations of the world. Only clothing that is immodest or extreme in style would be incompatible with wearing the garment.”
“The critical body of knowledge associated with the garment of the holy priesthood may be categorized under three headings: Armor of God, Historical Background, and Teachings of Modern Prophets.”
Armor of God: The garment, worn day and night, serves three important purposes: it is a reminder of the sacred covenants made with the Lord in His holy house, a protective covering for the body, and a symbol of the modesty of dress and living that should characterize the lives of all the humble followers of Christ. “…the white garment symbolizes purity and helps assure modesty, respect for the attributes of God, and, to the degree it is honored, a token of what Paul regarded as taking upon one the whole armor of God (Ephesians 6:13; cf. Doctrine and Covenants 27:15). … Garments bear several simple marks of orientation toward the gospel principles of obedience, truth, life, and discipleship in Christ.” The piece of armor called the temple garment not only provides the comfort and warmth of a cloth covering, it also strengthens the wearer to resist temptation, fend off evil influences, and stand firmly for the right.
Historical Background: “The scriptures contain many references to the wearing of special garments by the ancients. Prior to their expulsion from the Garden of Eden, Adam and Eve were clad in sacred clothing. We read: ‘Unto Adam also and to his wife did the Lord God make coats of skins, and clothed them’ (Genesis 3:21). The Book of Moses explains the sacred nature of those garments. Moses was commanded to place holy garments and priestly vestments upon Aaron and others, thus preparing them to officiate in the tabernacle. The honor, glory, and precious nature of sacred garments, whether worn only in the temple or in everyday life under street clothes, transcend the material of which they are made. Their full worth and beauty is appreciated and regarded as precious or glorious when viewed through the ‘eye of faith’” (Alma 5:15). Jews also continue to wear an undergarment, called a tzitzit, symbolizing holiness to the Lord and remembering the Mosaic commandments.
Teachings of Modern Prophets: “The principles of modesty and keeping the body appropriately covered are implicit in the covenant and should govern the nature of all clothing worn. Endowed members of the Church wear the garment as a reminder of the sacred covenants they have made with the Lord and also as a protection against temptation and evil. How it is worn is an outward expression of an inward commitment to follow the Savior.”
Much of the above information comes from the following article: Carlos E. Asay, “The Temple Garment: ‘An Outward Expression of an Inward Commitment,’ ” Ensign, Aug. 1997, 19.
 Boyd K. Packer, The Holy Temple(1980), 75.
 Evelyn T. Marshall, “Garments,” in Encyclopedia of Mormonism, ed. Daniel H. Ludlow 5 vols. (1992), 2:534; emphasis added.
 First Presidency Letter, 10 Oct. 1988; emphasis added.