One of the earliest businesses to expand at the beginning of the Industrial Revolution was the production of textiles. That growth could be accounted for by the simple but important factors of supply and demand -- Before the Revolution, the business, being a "cottage industry," meant that textiles were produced slowly, by hand, were expensive, and resulted in a limited supply. After the Revolution, were produced quickly, by machinery, and were cheap, and resulted in an abundant supply. The demand, high before the Revolution. became even higher afterwards, allowing for the business to expand production. However, other factors were critical in the process. The efficiency created by industrializing allowed for wealth to be created, which could be reinvested to improve the business. The growth of banking meant that additional capital became available to generate more capital, to invest in more businesses. Not only was money used to purchase better machinery, the machinery itself, the product of new inventions and sources of energy, were functions of more sophisticated technology, which also continued to be applied in innovative ways.
I assume you are asking why there was an increase in the size of firms during this time. And I'm assuming we're talking about the United States.
Factor 1: Technology. There is no one single piece of technology that did it, but there were technological breakthroughs in practically every large industry (example: the Bessemer Process for steel) that made it possible to produce goods faster and more efficiently. The greatest efficiency could be produced by large firms (economices of scale) and so large firms won out.
Factor 2: Increased competition between firms. This comes about in part because of railroads which allow companies in various parts of the country to compete with one another. Competition led to reduced and uncertain profits. Therefore, it was highly advantageous to combine small companies into larger ones that would control larger shares of the market.
The most important reason for growth was the development of technology, which introduced new and improved products as well as means to produce them economically in large quantities. Thus an opportunity was created for businesses to be set up and make profit.
The profit potential and growth of businesses was further enhanced by existence of large captive markets for the business in some European countries, particularly the Great Britain in form of colonies of these nations.
The trade in in larger geographical ares to sell larger quantities of products manufactured in factories was also aided by faster and cheaper means of transport, which were also the result of technological development.