This change didn't come from a team innovating play on the field, it stemmed from a rules change. The change in tackling wasn't so much a change from rugby style to head on as it was from group tackling to open field tackling. In early football the rules were fairly restrictive, formations were bunched up and the forward pass was illegal. As a result plays all worked pretty much like rugby plays, one man with the ball and all of his teammates would push one way, the other team would push back and when one side gave it was either a tackle or a short gain for the offense.
The problem with this style of play was people were getting killed. Some of the techniques used included players locking their arms and charging forward using their (helmet-less) heads as battering rams, or the "flying wedge" where the entire team would form a wedge formation and run straight into the other team. Protective gear was nonexistent, limited to a sweater and (sometimes) leather pants. No head face or rib protection. Sometimes ball carriers would be trampled underneath the entire team, crushed under the weight of thousands of pounds of other young men. At other times they would be battered senseless in the press. In 1905 with fatalities rising (2 deaths on the seasons final day brought the total to 19 for the year) President Roosevelt called on University Presidents to get a committee together to discuss uniform rules for eligibility and to clean up the brutality of the games. This led to the formation of the first major Intercollegiate athletic association, the IAAUS.
The biggest reforms involved making the forward pass legal and opening up formations. They also made it necessary to make 10 yards in 3 downs to keep drives going. It took quite a while for the forward pass to become popular, but the other reforms had a major and immediate impact. In 1906 the New York Daily Tribune hired Swarthmore Coach George Brooke to describe the tactical changes the new rules would force in a series of articles.
His best description was, "Beyond a doubt the game will be more open. There will certainly be a great deal more kicking, flukes, passing tricks, open field running and general hurry scurry."
Your damn right there was a lot more hurry scurry! The game changed pretty radically, and since offenses were playing a completely different style, hurry scurrying about from their open formations trying to tear off massive chunks of as much as 3 and 4 yards per play, the tackling had to change as well. Because the most efficient way to stop a runner from advancing in the open field is to shadow him laterally then stop him head on when he turns up field there was more head on open field tackling. A reflection of the new normal, 1906 style.
Here is Coach Brooke's original article on the new football tactics, courtesy of the Library of Congress: