What's the point of the transept?

An answer from French Wikipedia (perhaps not very different from what has already mentioned in the comments and the answer by @Luiz):

Outre le côté symbolique de la croix latine, le transept tient aussi un rôle technique de premier ordre en participant à la solidité de l'édifice, à la manière de deux arcs-boutants géants. Il augmente également l'espace intérieur de l'église et joue son rôle ordinaire de lieu de passage marquant la séparation liturgique entre le chœur des religieux à l'est et la nef des fidèles à l'ouest. Enfin, en démultipliant le nombre et la surface des murs, il permet, sans préjudice pour la stabilité de l'ensemble et grâce à ses fenêtres, d'apporter un éclairage latéral au sanctuaire ainsi que des portes supplémentaires qui régulent la circulation intérieure2.

In addition to the symbolic side of the Latin cross, the transept also plays a major technical role by contributing to the solidity of the building, like two giant flying buttresses. It also increases the interior space of the church and plays its ordinary role as a place of passage marking the liturgical separation between the choir of religious to the east and the nave of the faithful to the west. Finally, by increasing the number and surface area of the walls, it allows, without prejudice to the stability of the whole and thanks to its windows, to provide side lighting to the sanctuary as well as additional doors which regulate interior circulation.

The Answer in Italian Wikipedia is also of interest (as Italians also have some expertise in building Catholic Churches):

All'incrocio tra navata e transetto si trova spesso un elemento decorativo o strutturale, atto a sottolineare il ruolo architettonico svolto dallo stesso; non è raro, infatti, specie in età medievale, che l'altare maggiore si trovi proprio in tale corrispondenza. Esso può assumere le forme di una cupola (è il caso del Duomo di Milano), una torre (come nella Cattedrale di Salisbury) o di una guglia (come nella Cattedrale di Notre-Dame a Parigi).

L'orientamento della navata trasversale, cioè il transetto, è determinato da quello principale della navata centrale, che prevedeva originariamente la facciata a ovest e l'abside ad est, seguendo cioè il corso del sole. In questa configurazione, comune in Italia fino al XIII secolo, la chiesa veniva detta orientata.

Le potenzialità della struttura, che interrompe le serie strutturali della navata, generalmente articolate in cappelle laterali o navate minori illuminate da finestroni, vengono spesso sfruttate per favorire l'ingresso e l'uscita dei fedeli dall'edificio: ecco allora apparire, anche in corrispondenza del transetto, maestose facciate, spesso ricche di richiami ornamentali e architettonici alla facciata principale, e portali laterali, solitamente in corrispondenza delle singole navate in cui è suddivisa la navata trasversale.

At the intersection between the nave and the transept there is often a decorative or structural element, designed to underline the architectural role played by it; it is not uncommon, in fact, especially in the Middle Ages, for the main altar to be found precisely in this correspondence. It can take the form of a dome (as is the case of the Milan Cathedral), a tower (as in the Salisbury Cathedral) or a spire (as in the Notre-Dame Cathedral in Paris).

The orientation of the transverse nave, i.e. the transept, is determined by the main one of the central nave, which originally included the façade to the west and the apse to the east, i.e. following the course of the sun. In this configuration, common in Italy until the 13th century, the church was said to be oriented.

The potential of the structure, which interrupts the structural series of the nave, generally divided into side chapels or minor naves illuminated by large windows, is often exploited to facilitate the entry and exit of the faithful from the building: here it then appears, even in correspondence of the transept, majestic facades, often rich in ornamental and architectural references to the main facade, and lateral portals, usually in correspondence with the individual naves into which the transverse nave is divided.

Similarity to the Latin cross
The collection Le transept et ses espaces élevés dans l’église du Moyen Age (XIe-XVIe siècles). (The transept and its elevated spaces in the Middle Ages church (11th-16th centuries). Provides wealth of information on functional and esthetic role of the transept and, importantly, how these evolved over centuries. In particular, the introduction provides extended comment on the role of the cross-like shape (interestingly, term transept infiltrated French only in the XIX-th century - up to then the term was crux):

Au Moyen Âge, lorsqu’il s’agit de désigner cette partie de l’édifice, les exégètes de l’église emploient le plus sou- vent le mot « crux », de forte signification symbolique puisque les fidèles se voient ainsi rassemblés sous le signe du Christ crucifié, leur modèle. Lorsqu’elle est cruciforme, l’église reproduit plus exactement la forme du corps humain: le sanctuaire en est la tête, le transept les bras et les mains, la nef les membres inférieurs. Dans un édifice pensé en tant qu’équivalent matériel de l’Ecclesia spirituelle, rassemblant (physiquement et symboliquement) les fidèles, l’organisation des espaces correspond à celle de la société: le sanctuaire, soit l’emplacement de l’autel, est celui du clergé qui « prie, prêche, chante, et administre les choses saintes », tandis que la nef est réservée au peuple qui « écoute et qui prie ». L’utilisation du mot « crux » pour qualifier et désigner la structure architecturale souligne la valeur hautement symbolique de ce vaisseau pour les chrétiens, une structure qui apparaît dès le développement de la première architecture chrétienne au IVe siècle, à Saint-Pierre de Rome notam- ment mais également à Saint-Paul-hors-les-Murs. Soumise à une organisation longitudinale des espaces, l’église l’est aussi dans un rapport de latéralité: les hommes, « plus fermes dans la foi » et donc plus résistants aux assauts de la tentation, occupent la partie sud de l’édifice, la plus exposée, tandis que les femmes, plus faibles et moins résistantes aux tentations, doivent se tenir dans sa partie nord.

In the Middle Ages, when it came to designating this part of the building, church exegetes most often used winds the word “crux”, of strong symbolic meaning since the faithful see themselves gathered together under the sign of Christ crucified, their model. When it is cruciform, the church more exactly reproduces the shape of the human body: the sanctuary is the head, the transept the arms and hands, the nave the lower limbs. In a building designed as the material equivalent of the spiritual Ecclesia, bringing together (physically and symbolically) the faithful, the organization of spaces corresponds to that of society: the sanctuary, i.e. the location of the altar, is that of the clergy who “pray, preach, sing, and administer holy things", while the nave is reserved for the people who "listen and who pray.” The use of the word “crux” to qualify and designate the architectural structure underlines the highly symbolic value of this vessel for Christians, a structure which appears from the development of the first Christian architecture in the 4th century, at Saint-Pierre in Rome in particular ment but also in Saint-Paul-hors-les-Murs. Subject to a longitudinal organization of spaces, the church is also in a laterality relationship: men, “firmer in the faith” and therefore more resistant to the onslaught of temptation, occupy the southern part of the building, the most exposed, while the women, weaker and less resistant to temptations, must be held in its northern part.