As a first point, in general, the dotted line indicates that the nodes are in the same information set. So in your example, there is a single information set for the Red player and as such she has 2 actions---since she cannot tell which of the 3 nodes she is in, she cannot condition her action on this information, and therefore must take the same action (either left ($l$) or right ($r$)) irrespective of her position.
If, instead, there was no dotted line, then the Red player would know which action the Blue player had taken: lets call them $L$, $M$, or $R$. Contrary to your understanding, in this situation, the Red player does indeed have 8 strategies. Specifically, she can choose to play $l$ or $r$ conditional on each of the three actions she might observes Blue play: letting $(x,y,z) \in \{l,r\}^{3}$ denote the strategy where Red plays $x \in \{l,r\}$ conditional on $L$, $y \in \{l,r\}$ conditional on $M$, and $z \in \{l,r\}$ conditional on $R$, the 8 strategies are
$$(l,l,l),(l,l,r),(l,r,r),(l,r,l),(r,l,l),(r,l,r),(r,r,r),(r,r,l)$$
To understand more generally what is going on, it might help to think less formally and more philosophically about what an information set is. An information set is a collection of game positions (nodes) that a player cannot distinguish between (you can think of the dotted line as representing this entanglement). Then, if a player cannot distinguish between positions, she certainly cannot choose different actions at different positions, for how would she practically make this happen. So the player makes one choice per info set. If, on the other hand, the player can tell the difference between positions, she has many info sets and thus must make correspondingly many more choices.
The combinatorial explosion in the strategy space comes from the fact that each of these choices can be made independently.1 So changing what the Red player will do after the choice of $L$ by the Blue player yields two strategies for
every fixed choice out of the other information sets. When there are more periods of play, this growth becomes very very fast (I remember in grad school calculating the number of strategies in tic-tac-toe to be much larger than the atoms in the observable universe). On the other hand, having many strategies does not make the game necessarily more complicated, as many of the strategies may be identified via symmetry or other heuristics.
1 There is a subtly here that some actions might preclude reaching other information sets: nonetheless, in general a strategy is taken as a choice of action at every information set---including the ones precluded by earlier actions in the same strategy
