The Theory of Corporate Finance is an academic field that tries to explain theoretically why companies issue the kinds of securities they do. The simplest company, a steady state business that earns say X a year growing at y% a year could be financed with equity and (optionally) debt alone. How then do we explain the great variety of instruments that real life companies issue, including warrants, convertible debt (of different kinds) and so on?
TCF has been able to explain this to some extent by saying that companies are more complicated than described above, for example that they have "growth options" or "investment options" available to them for a limited period of time and that these "options" are not available to every company.
It is important to understand that these are not "financial options" like puts and calls traded on the CBOE, but "real options". In an abstract sense they are somewhat analogous, and will have a "premium" and a "maturity date" and so on. But they fairly abstract entities.
For example a mining company may have discovered gold on a property. By paying a "premium" of 500,000 to a geologist, they can have a 6 month study done to map the field and determine the feasibility of mining it. On or before the 6 month "maturity" they can decide, if everything looks good, if the price of gold is high etc. to "exercise" the growth option and pay 2,000,000 to build a mine on the site to produce gold. Another example would be a company that is developing a new product, and if successful, will spend money to build a factory to produce the product. Again we have an "option" that will either expire (the product does not work or is not worth selling) or "be exercised" by spending an "exercise price" to purchase a shiny new factory that will produce the product. These options (or "business opportunities", to use a simpler language) have a "maturity date" that could be years away in the case of an R&D program that is expected to last that long before failure or success.
What does this have to do with the existence of convertible bonds? The theory (and it is a theory, to be verified or rejected empirically) says that steady state companies do not issue convertibles, they are issued by "growth companies" companies with "growth options" or "investment options" available to them. These companies are not in great financial shape now, but they are able to attract money by offering the following deal: We will issue conv. bonds now, if things go well for us our equity will go up and you will able to convert to equity and make a lot of money. If things don't go well (the option is not exercised, the business opportunity does not pan out) you will still be bond holders and will have some protection in bankruptcy court if our equity is wiped out.
The article you mentioned "Convertible Bond Design and Capital Investment: The Role of Call Provisions" is part of this "real options" literature. It tries to prove that kind of "investment options" that a company has determines the kind of cb they issue. Companies that have investment options that "mature" in the distant future (for example a pharma company with a long term R&D program) will issue callable convertible bonds, while the others (for example a mining company with a 6 month deadline for opening/closing a mine) will issue regular (non-callable) convertible bonds).
Sorry for a complicated answer to a very simple question. The "maturity" we are concerned with here is the maturity of the "real options" or "business opportunities" available to the company. The time available until a decision is to be made whether to pursue the opportunity (by investing company money) or not.