First, to the facts. Organized hockey tournaments began with the 1883 Montreal Winter Carnival, which was won by McGill, and also featured only 1.13 goals scored per team per game. Such low scoring rates continued in the 1884 and 1885 Winter Carnivals, and the 1886 Montreal hockey championship (held due to the lack of a Carnival that year). In 1887, between the Winter Carnival games and challenge matches of the newly-formed Amateur Hockey Association of Canada (AHAC), over two goals were scored per game, and starting in 1888 is was substantially higher than that, generally approaching three per game. The data is here:
Something clearly changed around 1887, and as well already know, it's nothing to do with the roles players played on the ice. My hypothesis is that it is largely to do with something that can be seen by comparing portraits of the Montreal AAA hockey teams from 1885 and 1893. First the 1885 club:
And now the 1893 players. Take note of the long wooden items in their hands.
The sticks are noticeably longer. The 1885 sticks seem more like field hockey sticks, and required players to hunch over quite a bit when stickhandling. The 1893 sticks, while still short by modern standards, are a fair bit longer and the blades more closely resemble modern ones as well. I believe that the additional length of stick made stickhandling and shooting that much more accurate, and as such scoring increased all around.
By spending some time with some old hockey photos and a ruler, I came up with some estimates of the length of hockey sticks at various points in early history. We know that for the 1927/28 season, the NHL instituted a rule that sticks would be no longer than 53 inches, measured from the top of the handle to the heel of the blade. In earlier years, sticks were shorter than this, apparently increasing in length fairly steadily over time, except for a noticeable jump somewhere between 1885 and 1893.
In 1885, hockey sticks were about 33 to 35 inches, measured from the top of the handle to the heel of the blade.
In 1893, they were about 42 to 46 inches.
In 1901, they were about 44 to 49 inches.
In 1910, they were about 46 to 52 inches.
As an illustration of these differences, here are some badly-drawn hockey stick shapes: