The use of folded chasubles, in the liturgical books planetis plicatis, is an ancient characteristic of the Roman liturgy even if the 'modern' style of vestment seen has evolved somewhat from the ancient form. It seems probable that in the early Church secular dress, or something quite close to it was worn by the clergy or rather the development of distinct clerical attire came later. A famous image of St. Gregory with his parents, Gordianus and Silvia, was reproduced in Cyril Pocknee's 'Liturgical Vesture', Mowbrays, 1960 with the image taken from Rocca, 'S Gregorii, ejusdem parentum imagines', Rome, 1597. This image, photographed below, shows St. Gregory with his parents. All three are wearing the paenula (or chasuble) over a dalmatic underneath which is an alb. The only difference is that St. Gregory wears a pallium over his paenula and his mother's alb is decorated at its hem.
Liturgical historians believe that these very full chasubles were worn by all the clergy (and laity alike) during the early centuries. With the development of distinct liturgical vesture for the clergy and the later introduction of the dalmatic (which was regarded as a sign of joy) in penitential seasons such as Advent the older practice of all the clergy wearing chasubles was retained. As can be seen from the image arm movement for St. Gregory and his parents is somewhat restricted by the flowing folds of the chasubles. The generally accepted theory is that the sub-deacon and particularly the deacon found the full, beautiful, chasubles so restrictive when holding books to sing the pericopes and carry out other functions that they initially folded them up at the sides to allow easier movement or simply removed them. Then for convenience the deacon folded his chasuble up lengthways and threw it over his shoulder.
Over time, particularly since the seventeenth century, the chasuble has generally become a much reduced and cut-down vestment in its dimensions. As this style of chasuble doesn't restrict arm movement at all they became symbolically folded at the front - planetis plicatus ante pectus - as the liturgical books put it. With full vestments it is probably likely that the folding was at the sides but with the 'Roman' style the folding up became symbolic at the front but with many folded chasubles, like the St. Magnus set pictured, not really being folded at all but made with short fronts. [Update: I have been corrected - the St. Magnus set have been folded up and stiched!]
The rules for their use are these. On the ferial days and Sundays of Advent and Lent (except for Gaudete and Laetare Sundays), Ember tide (excepting the Pentecost Ember Days), the Vigils of Easter and Pentecost, and at Candlemass the deacon and sub-deacon wear folded violet chasubles. On Good Friday, uniquely, black folded chausubles are worn for the Mass of the Pre-Sanctifed. (At Pontifical functions on the same days the assistant deacons and canon deacons etc wear the same.) During the last collect the sub-deacon goes to the bench and removes his folded chasuble. He sings the Epistle and then, after receiving the celebrants blessing he resumes his chasuble and moves the missal as he usually does. When the celebrant begins to read the Gospel the deacon goes to the credence table and takes off his folded chasuble. Depending on the style of the vestment he may then either fold it lengthways and put it over his shoulder or, and much more likely with Roman style vestments, he puts over his diaconal stole a broad band of silk called the 'broad stole' which represents the folded chasuble. He then takes the Gospel book and serves the rest of Mass as he usually does but only wearing the broad stole. After Communion when he has moved the missal back to the Epistle side he takes off the broad stole (or unhitches his folded chausble) and put his folded chasuble back on.
( In the picture above from St. Magnus the Martyr the deacon can be seen to the celebrant's right wearing his broad stole. The sub-deacon wears the humeral veil as he usually does.)
(The photograph above is again from Orbis Catholicus report on Mass at the Pantheon. The ministers are leaving the altar. Note that although a penitential season the protonotary celebrant still wears his violet choir cassock (unlike a bishop outsise of Rome) as protonotaries and domestic prelates only wear black choir dress sedevacante.)
Recently Shawn Tribe at The New Liturgical Movement discovered some photographs showing folded chasubles that looked rather different and perhaps how they may have looked in the period before vestments were radically cut back. The photographs were taken at the Belgian Abbey of St. Andrew. (Origin of the popular and excellent Daily Missal). The article is well worth reading.
(The subdeacon. Note the effect of folding the semi-full chasuble at the front has on its sides)
(The deacon. Note how the chasuble has been folded up lengthways and compare with the photograph of the deacon in broad stole at St. Magnus.)
Folded chausubles became a casualty of new order of Holy Week published in 1955 (although were still used in the experimental Easter Vigil services in both its 1951 and 1952 forms) and then were thrown out of the rest of the liturgical year in 1961. One can only hope that much more is seen of folded chasubles in many more places in the future.