Mathematics of the income and substitution effects
The (Hicks-) substitution effect is by definition the change in consumption of X induced by a change of the relative prices, holding utility fixed. Thus, the original budget line is "rolled along" its indifference curve until it has the same slope (same relative prices) as the actual budget constraint after the price change. This determines $e'$. You can think of the tangent to $e'$ as a hypothetical budget constraint that compensates you for the lost utility resulting from the increasing price of $X$ by adding income until you can just afford a bundle with the same utility as the original bundle $e$. In total, the change from $e$ to $e^*$ is then separated into a change from $e$ to $e'$, (rolling the budget line along the indifference curve: relative price change, holding utility constant = substitution effect) and a change from $e'$ to $e^*$ (parallel shift of budget line: income change, holding relative prices constant = income effect).