This question deserves a longer answer than I can give right now (on a break from work), but I'll do my best to give you the short and sweet version.

There are two major schools of historical thought when it comes to this question. They are known by various names, but we'll call them the "social control" and "medical advancement" theories.

Both theories depart from the same starting point. Before the 19th century, there were few organized systems of care for the mentally ill. There were a few private asylums (the York Retreat in England, which was run by Quakers, comes to mind), but the state-funded infrastructure in Europe, Britain, and North America was non-existent. The massive demographic changes of the early and mid-century (which themselves are subject to a separate historical debate- was it declining infant mortality, increased wealth allowing more kids, etc.) led to a crisis point - there were more visible "pauper lunatics" than ever before.

Here's where the theories diverge. "Social control" theorists, beginning with Michel Foucault (History of Madness) and continuing in branching directions to the works of Thomas Szasz ("myth of mental illness"), David Rothman, Andrew Scull, etc. all posited variations of a thesis that placed hegemonic state control at the centre of mental health care (colloquially "lunacy care") expansion. Essentially, they suggest that a rapidly centralizing and bureaucratizing state implemented insane asylums as a "panoptic" method of class-based hegemonic control. "Pauper lunatics" (the impoverished mentally ill, or as some theorists would argue, non-mentally ill) were institutionalized to remove their aberrant, deviant, or non-contributory presence from society. These theories are most often supported by the evidence of the failure of insane asylums to cure mental illness by the end of the century. They essentially became custodial institutions as no medical therapies were found to be successful.

"Medical advancement" theorists (Edward Shorter comes to mind, A History of Psychiatry) posit that the explosion of institutionalization in the mid-century arose from actual advancements in mental health care. They point to the early twentieth-century breakthroughs in pharmacology and draw connections to earlier treatments.

A new generation of historians have begun to complicate this binary assessment. I'm running out of time, but suffice it to say that these two theories each have their advantages, and they may have been split more by the innate tendency of historians to resort to polemics than any actual incompatibility. I'd be happy to expand on this later, but I would suggest that the increase of asylums in general (public asylums didn't "arrive" in the US and Canada until the mid-1800s) and of admissions was due both to the structural changes experienced by society (immigration, population increase, state expansion, and yes, state control) and to benevolent impulses awakened by religious movements such as the Second Great Awakening. The advancement of medicine, as well as the eagerness of physicians to establish themselves within their chosen profession, led to the formation of associations (such as the Association of Medical Superintendents of American Institutions for the Insane) which lobbied and petitioned for lunacy care and its recognition in the medical community.

I hope this helps. I know it isn't a "patient-level" answer. I can't account for actual increases in mental illness (which is possible, and is suggested by some works of modern cultural history, which point to the prevalence of nervous diseases such as "neurasthenia") but maybe someone else here can.

Anyway, I hope this helped at least a little. Great question!

David Wright's 1998 paper Getting Out Of The Asylum: Understanding The Confinement Of The Insane In The 19th Century provides a good overview of the various factors that played a role in this; Wright's paper attempts to look for commonalities in a great deal of research on asylums in specific areas and times.

Firstly, mental asylums had existed for a long time before the 19th century. A good illustration of this is the English word 'bedlam', used to describe madness. The word is derived from the Bethlem Royal Hospital in London (which was primarily an institution for the mentally ill by about 1460). The English word has been in common use since at least Shakespeare (who has a character in Henry VI say "to Bedlam with him!").

However, the treatment of patients in institutions like Bethlem was primitive, with patients treated, basically, like wild animals. What was new in the 19th century was a new ideology of treatment, called 'moral treatment'. This was influenced by a mix of late 18th century physicians like Phillippe Pinel in France and religiously motivated types like William Tuke in York in England. Moral treatment involved treating the inmates at the asylum like gasp humans able to make moral choices, and originally involved putting a whole lot more structure in the inmates' lives rather than simply housing the mentally ill and treating them punitively.

As 'moral treatment' became more and more routine in the care of mentally ill patients, families having to care for aggressive or dependent people with mental illnesses began to see asylums as suitable for the family members that needed caring for; as a result, asylums expanded significantly. Between 1800 and 1860 various areas of the Western world passed legislation sometimes enabling and sometimes obliging local authorities to care for the mentally ill. However, the sheer demand for places in these asylums caused the system to become unworkable, and asylums again became, basically, housing for mentally ill people who were generally ignored.

According to Wright, the medical profession has previously systematically overemphasised its contribution to the rise of asylums in the 19th century, because of the rise in prominence of psychiatry in the 20th century. However, medical professionals in asylums in the early 19th century actually often complained of being sidelined and ignored, and having little power to choose inmates or how they were treated. Instead the role of such 'alienists' seemed to be almost peripheral to treatment, to rubber stamp the entry and exit of inmates. The majority of patients in many settings were placed there by families who were having difficulty dealing with mentally ill family members, and the average stay in an asylum was about a year, rather than the indefinite stay that could stretch on for decades that is the stereotype.

Wright seems to see the rapid rise of the asylums as being, basically, a transfer from informal and regional solutions to the problem to more formal and centralised solutions to the problem, as you'd expect in an industrialising society:

Before the construction of public asylums, poor insane persons were regularly admitted to local workhouses, gaols, or more rarely (by parochial authorities) to private madhouses, areas where they were often not reported in official statistics. When public asylums were created, there was an exodus of lunatics out of these 'welfare institutions' and into the asylum. The sudden increase in the recorded residential insane population is largely, therefore, a statistical artefact reflecting the shift from one type of custodial institution to another. Secondly, the rate of growth in the number of people resident in asylums, after taking into account the rapid population growth of society in general, was moderate.