Coase begins his discussion of the nature of the emergence of the firm in section II of his argument (link provided). He argues there are several reasons a firm may arise.
First, he claims that they may arise out of a natural order of humanity. Specifically, some people would prefer to work for another because it is easier to be told what to do than figure it out for themselves. This reduces the burden of entrepreneurship for those not naturally inclined. However, Coase argues this cannot be a main reason because there is the mantra of every man being his own boss.
Coase includes other situations where a firm might arise. A person may want to control others and therefore pay them in part for the privilege or a firm may exist because the products produced are superior to those produced by an individual. The latter idea has some merit, but not because of quality but quantity. One person may only produce ten widgets a day, where a firm can produce 1000 widgets a day. The increased production influences demand driving down cost and thereby increasing demand.
Ultimately Coase decides the price mechanism of production is the driving force behind the creation of firms. Contracts between individuals have a cost which must be mitigated and a firm is the application which mitigates. Multiple contracts between individuals require time to negotiate, time to manage, and other costs for the exchange of production. However, in a firm everyone is working under the direction of one entity and it doesn't cost anything to have A work with B. This eliminates cost and can lower price points.
The final conclusion is this; it is cheaper for people to work together in a firm under a single director than for a collection of individuals to produce a product.