How were concussions treated in late 19th-century America?

Merck's 1899 Manual of the Materia Medica, described as a "ready reference pocket book for the practicing physician" has the following:

Cerebral Concussion.

Rest: absolute to be enjoined. Stimulants to be avoided. Warmth: to extremities.

A Manual of surgical treatment v.7, 1899-1903 (published in 1901) goes in to much more detail:

In the treatment of a patient in a state of concussion two things must be borne in mind. In the first place measures must be taken to get rid of the shock, while in the second place, great care must be taken to avoid bringing about too much reaction lest haemorrhage should occur. The patient should be placed in the recumbent position, with a very small pillow beneath the head, which must not be too much depressed on account of the risk of increasing any haemorrhage that may have begun. Warmth should be applied to the feet and limbs by means of hot blankets and hot-water bottles. Strychnine (gr. 1/25th) may be administered subcutaneously and also camphor, and in bad cases it may be advisable to put the patient into a warm bath. The administration of alcoholic stimulants, or of ether should be avoided in all but desperate cases, on account of the great risk of increasing the reaction and bringing about internal haemorrhage. The patient should be placed in a room free from noises and bystanders and a brisk purge should be administered while he is still unconscious (five grains of calomel placed on the back of the tongue or one or two drops of croton oil on sugar).

Also mentioned is that, in cases of concussion,

Retention of urine is extremely common, and a soft catheter should always be passed soon after the injury.

There is more in this source which may be of interest to you, including more serious cases of concussion involving contusion and laceration of the brain (treatment was more of the above).

Concussion in American Football, 1890s & 1900s

Concussion (and the number of injuries in general) was considered a serious issue in American Football in the 1890s and early 1900s by many colleges. A 1906 article in The Boston Medical and Surgical Journal, The Physical Aspect of American Football has this on the treatment of players suffering from concussion (referring to the 1905 season):

Players who had had concussion were at once carefully examined to exclude the possibility of middle meningeal hemorrhage and, during the earlier part of the season, were sent to their rooms or to their homes, with a companion, with strict orders that they were to be left alone at no time until noon of the following day.

How seriously it was considered by the general population would probably have depended on the individual and the way he/she was behaving, but

Concussion was treated by the players in general as a trivial injury and rather regarded as a joke.

This was despite the fact that

The mental state of the players who had concussion was variable, some being highly excitable and hysterical, others merely confused, and in a few cases, knocked completely unconscious. In every case there was a certain loss of memory, both previous and subsequent to the injury. The loss of memory previous to the injury varied from a few minutes to a week. In all cases also there was a loss of memory as to facts occurring for a variable time subsequent to the injury. For instance, it was common to hear a player ask if he had played the first or second half of the game.

There was, among those who sort to keep American football as they then were, an apparent tendency to downplay concussion to the point of censorship. For example, an response to an 1894 questionnaire "to players, coaches, and other aficionados", one William Harvey detailed his head injury in his letter but:

On his original letter, preserved in Camp’s papers at Yale University, Harvey’s description of his head injury has been blatantly crossed out in crayon.

As reported in the link provided by Pieter Geerkens, by 1905 the situation had prompted President Teddy Roosevelt (whose son was among the injured) to get involved. Those for and against the then current situation being allowed to continue were summoned to the White House as

In 1905 alone, at least 18 people died and more than 150 were injured playing football. According to the Washington Post, at least 45 football players died from 1900 to October 1905, many from internal injuries, broken necks, concussions or broken backs.

A number of rule changes soon followed, but helmets were not required until 1939.

Historical Note

The authors of the above cited article, The First Concussion Crisis: Head Injury and Evidence in Early American Football provide a little historical background on concussion:

The concept of brain concussion had been traveling through common and medical usage for centuries. As early as the mid-16th century it had been defined as a blow resulting in escape of blood from ruptured tissue. By the early 19th century it was described as an “external violence” that caused “derangement of the organization of the brain."

The turn of the century was a time of heightened awareness to injury...

some physicians expressed frustration that their peers paid too little attention to head injuries. “We all know that any impairment of the brain seriously affects the entire economy,” wrote one....Treating observable injuries like fractures with rest, symptom management, and reduction of stimulation might, he postulated, also care for invisible injuries in inaccessible parts of the brain.

Physicians around the turn of the century were hampered by increasing demands in the medical profession for proof of an injury rather than just clinical observation. Such proof was difficult to obtain for head injuries due to limited medical technology at the time. Thus,

There was no agreed mechanism, no theoretical consensus, for how concussion injured the brain.