Jesus Christ is the ultimate minister of ALL Sacraments.
The difference between Western and Eastern viewpoints concerns who is the proximate (earthly) minister. In Western theology, the bride and groom are the proximate ministers of the Sacrament, mutually conferring it upon each other (the ordained minister – at least a deacon – only pronounces the blessing of the Church upon the union). In the Eastern view, the bishop/priest (and perhaps a deacon – not sure) is the proximate minister. However, the Eastern minister must obtain the consent of the bride and groom before proceeding (a requirement not technically present in any other Sacrament).
You ask, “How did the Western understanding develop?” It was actually the original understanding. The Eastern understanding was not developed until about 1560. Per the Catholic Encyclopedia:
Although the Church realized from the first the complete sacramentality of Christian marriage, yet for a time there was some uncertainty as to what in the marriage contract is the real essence of the sacrament; as to its matter and form, and its minister. From the earliest times this fundamental proposition has been upheld: Matrimonium facit consensus, i.e. Marriage is contracted through the mutual, expressed consent. Therein is contained implicitly the doctrine that the persons contracting marriage are themselves the agents or ministers of the sacrament. However, it has been likewise emphasized that marriage must be contracted with the blessing of the priest and the approbation of the Church, for otherwise it would be a source not of Divine grace, but of malediction. Hence it might easily be inferred that the sacerdotal blessing is the grace-giving element, or form of the sacrament, and that the priest is the minister. But this is a false conclusion. The first theologian to designate clearly and distinctly the priest as the minister of the Sacrament and his blessing as the sacramental form was apparently Melchior Canus (d. 1560).
See the article I cited for Melchior’s reasoning and the arguments against it.